| L(s) = 1 | + 2·5-s − 5·9-s − 11-s − 8·13-s − 4·16-s + 4·17-s − 7·25-s + 7·31-s + 12·43-s − 10·45-s + 16·47-s − 10·49-s − 2·55-s + 10·59-s − 24·61-s − 16·65-s − 14·67-s − 6·71-s − 8·73-s + 20·79-s − 8·80-s + 16·81-s + 12·83-s + 8·85-s − 14·97-s + 5·99-s − 32·103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 5/3·9-s − 0.301·11-s − 2.21·13-s − 16-s + 0.970·17-s − 7/5·25-s + 1.25·31-s + 1.82·43-s − 1.49·45-s + 2.33·47-s − 1.42·49-s − 0.269·55-s + 1.30·59-s − 3.07·61-s − 1.98·65-s − 1.71·67-s − 0.712·71-s − 0.936·73-s + 2.25·79-s − 0.894·80-s + 16/9·81-s + 1.31·83-s + 0.867·85-s − 1.42·97-s + 0.502·99-s − 3.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1279091 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1279091 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9219645035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9219645035\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( 1 + T \) |
| 31 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 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
−7.78789607043645765017922373874, −7.64253009939598742504659377561, −7.32673252098760952172888757092, −6.47131284244476668201781833493, −6.23870064789214498217427813253, −5.72853996902860634151298445470, −5.41175071356617719099292784896, −5.01313827515890142438043543675, −4.44574490854809981150981932393, −3.96982013211312892858843970115, −3.00586813570449276279847345753, −2.63044898935838963010520851422, −2.44238240256814752006876582875, −1.65844747702205097833410469961, −0.39464352166213344612384621998,
0.39464352166213344612384621998, 1.65844747702205097833410469961, 2.44238240256814752006876582875, 2.63044898935838963010520851422, 3.00586813570449276279847345753, 3.96982013211312892858843970115, 4.44574490854809981150981932393, 5.01313827515890142438043543675, 5.41175071356617719099292784896, 5.72853996902860634151298445470, 6.23870064789214498217427813253, 6.47131284244476668201781833493, 7.32673252098760952172888757092, 7.64253009939598742504659377561, 7.78789607043645765017922373874
