| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 2·11-s − 12-s + 2·13-s − 14-s + 15-s + 16-s + 6·17-s − 18-s + 4·19-s − 20-s − 21-s − 2·22-s − 2·23-s + 24-s + 25-s − 2·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.426·22-s − 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.916133719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.916133719\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.45868662350104, −12.13991609954580, −11.79587548604993, −11.24170592879526, −11.05563537166229, −10.36174466805717, −9.976383671085109, −9.606992831956192, −9.042823593862920, −8.512774645480377, −8.002246341137557, −7.703847854215915, −7.204444351303218, −6.682200396500336, −5.978453597013513, −5.912781629410230, −5.126655732881466, −4.595849738122165, −4.094064272919465, −3.388630683711564, −3.039820198860180, −2.260595560096296, −1.314833660806339, −1.179874599091781, −0.5048108542876085,
0.5048108542876085, 1.179874599091781, 1.314833660806339, 2.260595560096296, 3.039820198860180, 3.388630683711564, 4.094064272919465, 4.595849738122165, 5.126655732881466, 5.912781629410230, 5.978453597013513, 6.682200396500336, 7.204444351303218, 7.703847854215915, 8.002246341137557, 8.512774645480377, 9.042823593862920, 9.606992831956192, 9.976383671085109, 10.36174466805717, 11.05563537166229, 11.24170592879526, 11.79587548604993, 12.13991609954580, 12.45868662350104
