| L(s) = 1 | + 7·4-s − 18·9-s + 124·11-s + 16-s + 244·19-s − 704·29-s + 132·31-s − 126·36-s + 32·41-s + 868·44-s − 98·49-s − 1.03e3·59-s − 1.76e3·61-s + 119·64-s + 620·71-s + 1.70e3·76-s − 3.66e3·79-s + 243·81-s − 1.59e3·89-s − 2.23e3·99-s + 3.10e3·101-s − 4.09e3·109-s − 4.92e3·116-s + 4.36e3·121-s + 924·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 7/8·4-s − 2/3·9-s + 3.39·11-s + 1/64·16-s + 2.94·19-s − 4.50·29-s + 0.764·31-s − 0.583·36-s + 0.121·41-s + 2.97·44-s − 2/7·49-s − 2.27·59-s − 3.69·61-s + 0.232·64-s + 1.03·71-s + 2.57·76-s − 5.21·79-s + 1/3·81-s − 1.89·89-s − 2.26·99-s + 3.05·101-s − 3.59·109-s − 3.94·116-s + 3.28·121-s + 0.669·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.121110804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.121110804\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 7 T^{2} + 3 p^{4} T^{4} - 7 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 62 T + 3582 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 6720 T^{2} + 20906318 T^{4} - 6720 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 15900 T^{2} + 109116038 T^{4} - 15900 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 122 T + 17398 T^{2} - 122 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1308 T^{2} + 549062 p^{2} T^{4} - 1308 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 352 T + 78278 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 66 T + 45870 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 53740 T^{2} + 3534883318 T^{4} - 53740 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 16 T + 18350 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 34620 T^{2} - 1693498 p^{2} T^{4} - 34620 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4180 T^{2} + 14462189158 T^{4} + 4180 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44384 T^{2} + 11570351278 T^{4} - 44384 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 516 T + 418118 T^{2} + 516 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 880 T + 575238 T^{2} + 880 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 74012 T^{2} + 114756705078 T^{4} - 74012 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 310 T + 664038 T^{2} - 310 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 1473328 T^{2} + 843769310398 T^{4} - 1473328 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 1832 T + 1824478 T^{2} + 1832 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1027340 T^{2} + 679923446038 T^{4} - 1027340 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 796 T + 411158 T^{2} + 796 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1841920 T^{2} + 2158510258558 T^{4} - 1841920 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.32096340095898246774710912719, −7.25097390259322107168228324747, −7.10623925716719638895432003291, −6.61047134681383072474328247354, −6.46760973555944416210231163172, −6.21225056256950016229078713362, −5.78314586977567984563206079364, −5.75731937302738976463607514912, −5.74711471816309571751595919773, −5.24277622308417801666530011906, −4.91664470153598504592470692001, −4.60491514471741011187902883542, −4.12312531176882143638939626380, −4.10584405886855613180340072901, −3.76250317392425434015619871047, −3.54077301345161542454763528334, −2.98568718307386687278641873315, −2.93953450709289900493870022123, −2.90805334431274191067107390342, −1.94162757884440659143311000644, −1.66175388036912982330645776556, −1.53153012872217405322416020043, −1.40047953370695651041519233341, −0.798245066365011784058325706290, −0.22118455616605298102314122836,
0.22118455616605298102314122836, 0.798245066365011784058325706290, 1.40047953370695651041519233341, 1.53153012872217405322416020043, 1.66175388036912982330645776556, 1.94162757884440659143311000644, 2.90805334431274191067107390342, 2.93953450709289900493870022123, 2.98568718307386687278641873315, 3.54077301345161542454763528334, 3.76250317392425434015619871047, 4.10584405886855613180340072901, 4.12312531176882143638939626380, 4.60491514471741011187902883542, 4.91664470153598504592470692001, 5.24277622308417801666530011906, 5.74711471816309571751595919773, 5.75731937302738976463607514912, 5.78314586977567984563206079364, 6.21225056256950016229078713362, 6.46760973555944416210231163172, 6.61047134681383072474328247354, 7.10623925716719638895432003291, 7.25097390259322107168228324747, 7.32096340095898246774710912719
