| L(s) = 1 | − 4-s + 9-s + 16-s + 2·19-s − 31-s − 36-s + 2·41-s − 49-s − 2·59-s − 64-s − 2·71-s − 2·76-s + 81-s − 2·101-s + 2·109-s + ⋯ |
| L(s) = 1 | − 4-s + 9-s + 16-s + 2·19-s − 31-s − 36-s + 2·41-s − 49-s − 2·59-s − 64-s − 2·71-s − 2·76-s + 81-s − 2·101-s + 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8601852130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8601852130\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.31612093946658127259255558600, −9.541611021326968732479624752316, −9.085181641310256509398098267720, −7.81881361926206685741979837580, −7.32176227966105889816011844902, −5.96346535895264340008388180143, −5.04498077952826148969029517906, −4.19292349190697612429970149170, −3.17798117511538719524909227738, −1.31769369520299349910855832753,
1.31769369520299349910855832753, 3.17798117511538719524909227738, 4.19292349190697612429970149170, 5.04498077952826148969029517906, 5.96346535895264340008388180143, 7.32176227966105889816011844902, 7.81881361926206685741979837580, 9.085181641310256509398098267720, 9.541611021326968732479624752316, 10.31612093946658127259255558600
