| L(s) = 1 | + 9-s + 4·11-s + 8·19-s − 18·29-s + 16·31-s − 24·41-s + 5·49-s − 24·59-s + 26·61-s − 12·71-s + 2·79-s − 12·81-s − 6·89-s + 4·99-s − 42·101-s + 10·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 1.20·11-s + 1.83·19-s − 3.34·29-s + 2.87·31-s − 3.74·41-s + 5/7·49-s − 3.12·59-s + 3.32·61-s − 1.42·71-s + 0.225·79-s − 4/3·81-s − 0.635·89-s + 0.402·99-s − 4.17·101-s + 0.957·109-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.84·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.845711648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.845711648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - T^{2} + 13 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 5 T^{2} - 27 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 53 T^{2} + 1233 T^{4} - 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 77 T^{2} + 2493 T^{4} - 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 57 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 4803 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 74 T^{2} + 2763 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 77 T^{2} + 3273 T^{4} - 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 133 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 13 T + 159 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T - 38 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 221 T^{2} + 22233 T^{4} - 221 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - T - 99 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 65 T^{2} + 13653 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 3 T + 175 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 233 T^{2} + 30873 T^{4} - 233 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−5.93818089134249958150696063022, −5.63662770458009889920042587895, −5.50734546682872376662024379502, −5.39339243518740575403218658570, −5.08403169885113036683159514632, −4.91397410411789914065325994907, −4.86301513005107134008589959840, −4.38482626561072370250400755569, −4.23830960898633452174474143019, −4.16248956289390071705329839527, −4.03745132899992785062877227856, −3.46487061229670831222301342209, −3.42232588254245779406412729190, −3.41001153887090481038786686783, −3.19380512017666953124710520274, −2.77394191210548824842656234159, −2.63469734103097501913416967577, −2.30370194219875638942630951753, −2.04586758329239130557493393719, −1.67458227338111246993113538677, −1.47798096105292030401073893233, −1.30514125433413281443499336127, −1.19396089585866482458430720296, −0.51120410345282507646669023921, −0.32653171371801238870142165391,
0.32653171371801238870142165391, 0.51120410345282507646669023921, 1.19396089585866482458430720296, 1.30514125433413281443499336127, 1.47798096105292030401073893233, 1.67458227338111246993113538677, 2.04586758329239130557493393719, 2.30370194219875638942630951753, 2.63469734103097501913416967577, 2.77394191210548824842656234159, 3.19380512017666953124710520274, 3.41001153887090481038786686783, 3.42232588254245779406412729190, 3.46487061229670831222301342209, 4.03745132899992785062877227856, 4.16248956289390071705329839527, 4.23830960898633452174474143019, 4.38482626561072370250400755569, 4.86301513005107134008589959840, 4.91397410411789914065325994907, 5.08403169885113036683159514632, 5.39339243518740575403218658570, 5.50734546682872376662024379502, 5.63662770458009889920042587895, 5.93818089134249958150696063022
