| L(s) = 1 | − 2-s + 4-s − 8-s − 4·11-s − 13-s + 16-s + 2·17-s − 4·19-s + 4·22-s − 2·23-s + 26-s + 6·29-s + 4·31-s − 32-s − 2·34-s − 4·37-s + 4·38-s + 6·41-s + 4·43-s − 4·44-s + 2·46-s − 52-s − 2·53-s − 6·58-s + 10·59-s − 2·61-s − 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.20·11-s − 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.852·22-s − 0.417·23-s + 0.196·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 0.342·34-s − 0.657·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s − 0.603·44-s + 0.294·46-s − 0.138·52-s − 0.274·53-s − 0.787·58-s + 1.30·59-s − 0.256·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.470366027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.470366027\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.63276653022104, −12.25402380487178, −11.85270212166824, −11.18387436736469, −10.80128354093133, −10.38767946272285, −9.969754250366844, −9.669349958024191, −8.918194076531434, −8.533678397542799, −8.135303660167634, −7.655675435914892, −7.282344349249293, −6.661106401061037, −6.147615502049621, −5.769671841816204, −5.046436960231960, −4.716231835606923, −4.027874726074780, −3.385943433184402, −2.758055957933263, −2.344191014262884, −1.832091416324361, −0.9094098414805286, −0.4386787434058973,
0.4386787434058973, 0.9094098414805286, 1.832091416324361, 2.344191014262884, 2.758055957933263, 3.385943433184402, 4.027874726074780, 4.716231835606923, 5.046436960231960, 5.769671841816204, 6.147615502049621, 6.661106401061037, 7.282344349249293, 7.655675435914892, 8.135303660167634, 8.533678397542799, 8.918194076531434, 9.669349958024191, 9.969754250366844, 10.38767946272285, 10.80128354093133, 11.18387436736469, 11.85270212166824, 12.25402380487178, 12.63276653022104
