| L(s) = 1 | + 17·4-s − 27·9-s + 66·11-s + 74·16-s + 338·19-s + 554·29-s − 346·31-s − 459·36-s + 88·41-s + 1.12e3·44-s + 1.45e3·49-s + 1.36e3·59-s − 2.07e3·61-s − 230·64-s − 2.91e3·71-s + 5.74e3·76-s + 1.01e3·79-s + 486·81-s + 1.21e3·89-s − 1.78e3·99-s + 1.62e3·101-s + 26·109-s + 9.41e3·116-s + 2.54e3·121-s − 5.88e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 17/8·4-s − 9-s + 1.80·11-s + 1.15·16-s + 4.08·19-s + 3.54·29-s − 2.00·31-s − 2.12·36-s + 0.335·41-s + 3.84·44-s + 4.24·49-s + 3.01·59-s − 4.35·61-s − 0.449·64-s − 4.87·71-s + 8.67·76-s + 1.44·79-s + 2/3·81-s + 1.44·89-s − 1.80·99-s + 1.60·101-s + 0.0228·109-s + 7.53·116-s + 1.90·121-s − 4.25·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{12} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(28.54683742\) |
| \(L(\frac12)\) |
\(\approx\) |
\(28.54683742\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 + p^{2} T^{2} )^{3} \) |
| 5 | \( 1 \) |
| 11 | \( ( 1 - p T )^{6} \) |
| good | 2 | \( 1 - 17 T^{2} + 215 T^{4} - 2167 T^{6} + 215 p^{6} T^{8} - 17 p^{12} T^{10} + p^{18} T^{12} \) |
| 7 | \( 1 - 208 p T^{2} + 963152 T^{4} - 398805322 T^{6} + 963152 p^{6} T^{8} - 208 p^{13} T^{10} + p^{18} T^{12} \) |
| 13 | \( 1 - 11717 T^{2} + 60050355 T^{4} - 171934452414 T^{6} + 60050355 p^{6} T^{8} - 11717 p^{12} T^{10} + p^{18} T^{12} \) |
| 17 | \( 1 - 27504 T^{2} + 323484856 T^{4} - 2087973188642 T^{6} + 323484856 p^{6} T^{8} - 27504 p^{12} T^{10} + p^{18} T^{12} \) |
| 19 | \( ( 1 - 169 T + 28098 T^{2} - 2360147 T^{3} + 28098 p^{3} T^{4} - 169 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 23 | \( 1 - 59863 T^{2} + 1598203822 T^{4} - 24767623270739 T^{6} + 1598203822 p^{6} T^{8} - 59863 p^{12} T^{10} + p^{18} T^{12} \) |
| 29 | \( ( 1 - 277 T + 68543 T^{2} - 12078682 T^{3} + 68543 p^{3} T^{4} - 277 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 31 | \( ( 1 + 173 T + 42261 T^{2} + 6587230 T^{3} + 42261 p^{3} T^{4} + 173 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 37 | \( 1 - 104 p T^{2} + 1479419160 T^{4} - 160607293323366 T^{6} + 1479419160 p^{6} T^{8} - 104 p^{13} T^{10} + p^{18} T^{12} \) |
| 41 | \( ( 1 - 44 T + 133902 T^{2} - 3107666 T^{3} + 133902 p^{3} T^{4} - 44 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 43 | \( 1 - 445865 T^{2} + 85078538003 T^{4} - 8896779561490358 T^{6} + 85078538003 p^{6} T^{8} - 445865 p^{12} T^{10} + p^{18} T^{12} \) |
| 47 | \( 1 - 343208 T^{2} + 58105531160 T^{4} - 6892336754510414 T^{6} + 58105531160 p^{6} T^{8} - 343208 p^{12} T^{10} + p^{18} T^{12} \) |
| 53 | \( 1 - 749254 T^{2} + 253429789783 T^{4} - 48743413448625812 T^{6} + 253429789783 p^{6} T^{8} - 749254 p^{12} T^{10} + p^{18} T^{12} \) |
| 59 | \( ( 1 - 684 T + 535726 T^{2} - 196887298 T^{3} + 535726 p^{3} T^{4} - 684 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 61 | \( ( 1 + 1038 T + 631795 T^{2} + 333327140 T^{3} + 631795 p^{3} T^{4} + 1038 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 67 | \( 1 - 1611126 T^{2} + 1127396795607 T^{4} - 441089292520974292 T^{6} + 1127396795607 p^{6} T^{8} - 1611126 p^{12} T^{10} + p^{18} T^{12} \) |
| 71 | \( ( 1 + 1459 T + 1463380 T^{2} + 997438583 T^{3} + 1463380 p^{3} T^{4} + 1459 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 73 | \( 1 - 1418330 T^{2} + 1061027771295 T^{4} - 509021917955073900 T^{6} + 1061027771295 p^{6} T^{8} - 1418330 p^{12} T^{10} + p^{18} T^{12} \) |
| 79 | \( ( 1 - 506 T + 1295660 T^{2} - 507308816 T^{3} + 1295660 p^{3} T^{4} - 506 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 83 | \( 1 - 3078185 T^{2} + 4115985989371 T^{4} - 3068003031096076534 T^{6} + 4115985989371 p^{6} T^{8} - 3078185 p^{12} T^{10} + p^{18} T^{12} \) |
| 89 | \( ( 1 - 607 T + 1367067 T^{2} - 593095898 T^{3} + 1367067 p^{3} T^{4} - 607 p^{6} T^{5} + p^{9} T^{6} )^{2} \) |
| 97 | \( 1 - 4697939 T^{2} + 9794432387526 T^{4} - 11572077128936580567 T^{6} + 9794432387526 p^{6} T^{8} - 4697939 p^{12} T^{10} + p^{18} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.98790236595588346133155718159, −4.88798085282220026819920151983, −4.70347844369199870351734376640, −4.52412923664882846357759893057, −4.33240884004372481695020884301, −4.02480459051119246771829423014, −3.93532243788376720109143808828, −3.88538186308967808003943169684, −3.43460880633942993475535888588, −3.34303868774747218845094408911, −3.27066285531628959055683253862, −2.92570262279310880952907456440, −2.92344882352631442745012633099, −2.56703693922827706083367213764, −2.56374454620790454541480667019, −2.52093769019070613527859655370, −1.93704146998362575344802419368, −1.89877290027218116958942374658, −1.69050342382578813827057728800, −1.36581903008308745883066158917, −1.08107589526650009528952651053, −1.04741049333114436465454450466, −0.879310619942989634262106537295, −0.51473467988473173932800239605, −0.29566007449265509861407278872,
0.29566007449265509861407278872, 0.51473467988473173932800239605, 0.879310619942989634262106537295, 1.04741049333114436465454450466, 1.08107589526650009528952651053, 1.36581903008308745883066158917, 1.69050342382578813827057728800, 1.89877290027218116958942374658, 1.93704146998362575344802419368, 2.52093769019070613527859655370, 2.56374454620790454541480667019, 2.56703693922827706083367213764, 2.92344882352631442745012633099, 2.92570262279310880952907456440, 3.27066285531628959055683253862, 3.34303868774747218845094408911, 3.43460880633942993475535888588, 3.88538186308967808003943169684, 3.93532243788376720109143808828, 4.02480459051119246771829423014, 4.33240884004372481695020884301, 4.52412923664882846357759893057, 4.70347844369199870351734376640, 4.88798085282220026819920151983, 4.98790236595588346133155718159
Plot not available for L-functions of degree greater than 10.
