| L(s) = 1 | + 2-s − 4-s + 2·5-s − 7-s − 3·8-s + 5·9-s + 2·10-s − 14-s − 16-s + 5·18-s − 2·20-s − 25-s + 28-s + 5·29-s + 5·32-s − 2·35-s − 5·36-s − 6·40-s + 10·45-s + 10·47-s − 6·49-s − 50-s + 3·56-s + 5·58-s + 5·61-s − 5·63-s + 7·64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s − 0.377·7-s − 1.06·8-s + 5/3·9-s + 0.632·10-s − 0.267·14-s − 1/4·16-s + 1.17·18-s − 0.447·20-s − 1/5·25-s + 0.188·28-s + 0.928·29-s + 0.883·32-s − 0.338·35-s − 5/6·36-s − 0.948·40-s + 1.49·45-s + 1.45·47-s − 6/7·49-s − 0.141·50-s + 0.400·56-s + 0.656·58-s + 0.640·61-s − 0.629·63-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.074138666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.074138666\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 39 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.278877274892695287727795374037, −7.67079707009814488610000195947, −7.29485144321340666053514802234, −6.68430673500856150434857031881, −6.43964864815821305389806804844, −5.95604872381600692570630303817, −5.40825241828452335274084146043, −5.07343634313703593101845722614, −4.38561837143467013791385821649, −4.20700872251592628318421041853, −3.59483748815996456256666405091, −2.99169022499410819952197063053, −2.32440195501426568619157398366, −1.63852829411733375635312587819, −0.790582477150764616582705568130,
0.790582477150764616582705568130, 1.63852829411733375635312587819, 2.32440195501426568619157398366, 2.99169022499410819952197063053, 3.59483748815996456256666405091, 4.20700872251592628318421041853, 4.38561837143467013791385821649, 5.07343634313703593101845722614, 5.40825241828452335274084146043, 5.95604872381600692570630303817, 6.43964864815821305389806804844, 6.68430673500856150434857031881, 7.29485144321340666053514802234, 7.67079707009814488610000195947, 8.278877274892695287727795374037
