| L(s) = 1 | + 2-s − 4-s + 2·5-s − 3·8-s + 2·10-s − 13-s − 16-s − 2·20-s − 25-s − 26-s − 12·29-s + 5·32-s + 4·37-s − 6·40-s − 10·41-s + 6·49-s − 50-s + 52-s + 12·53-s − 12·58-s + 7·64-s − 2·65-s + 8·73-s + 4·74-s − 2·80-s − 9·81-s − 10·82-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.06·8-s + 0.632·10-s − 0.277·13-s − 1/4·16-s − 0.447·20-s − 1/5·25-s − 0.196·26-s − 2.22·29-s + 0.883·32-s + 0.657·37-s − 0.948·40-s − 1.56·41-s + 6/7·49-s − 0.141·50-s + 0.138·52-s + 1.64·53-s − 1.57·58-s + 7/8·64-s − 0.248·65-s + 0.936·73-s + 0.464·74-s − 0.223·80-s − 81-s − 1.10·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.089395280\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.089395280\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.26979981317250257589332515048, −11.79949554432528869309588984713, −11.17198916644455848485277943453, −10.32512773440289252367629441290, −9.861284303570639666430214160996, −9.246510314099117027026882899793, −8.833765465339395767129140532864, −7.971611496084204229317672988323, −7.19834695053174596084552775782, −6.35755945750334048321637775121, −5.62451786253477867098229694168, −5.25871869691954988315552418768, −4.24392083950085377986568225857, −3.44841223629915606080476063504, −2.18702148199447707048437796311,
2.18702148199447707048437796311, 3.44841223629915606080476063504, 4.24392083950085377986568225857, 5.25871869691954988315552418768, 5.62451786253477867098229694168, 6.35755945750334048321637775121, 7.19834695053174596084552775782, 7.971611496084204229317672988323, 8.833765465339395767129140532864, 9.246510314099117027026882899793, 9.861284303570639666430214160996, 10.32512773440289252367629441290, 11.17198916644455848485277943453, 11.79949554432528869309588984713, 12.26979981317250257589332515048
