| L(s) = 1 | − 2-s + 4-s − 8-s + 4·11-s + 13-s + 16-s − 2·17-s + 8·19-s − 4·22-s − 26-s − 6·29-s + 4·31-s − 32-s + 2·34-s + 2·37-s − 8·38-s − 10·41-s − 4·43-s + 4·44-s − 8·47-s + 52-s − 10·53-s + 6·58-s + 4·59-s + 2·61-s − 4·62-s + 64-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 + 1.83·19-s − 0.852·22-s − 0.196·26-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.342·34-s + 0.328·37-s − 1.29·38-s − 1.56·41-s − 0.609·43-s + 0.603·44-s − 1.16·47-s + 0.138·52-s − 1.37·53-s + 0.787·58-s + 0.520·59-s + 0.256·61-s − 0.508·62-s + 1/8·64-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
| \(L(1)\) |
\(\approx\) |
\(2.264768841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.264768841\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.76157833472253, −12.03944738494451, −11.63837354465346, −11.40004703232213, −11.07156763947735, −10.18712179815602, −9.977687847419061, −9.426860444505082, −9.141776769368535, −8.643964519871665, −7.976809474033402, −7.815477564096265, −7.056615650136997, −6.579375311499103, −6.453917546091500, −5.611958857013744, −5.181763559119572, −4.646742716817499, −3.885906800361428, −3.378164263919682, −3.107432433370288, −2.089210378881618, −1.735427086029324, −1.053488151721106, −0.5115085972324189,
0.5115085972324189, 1.053488151721106, 1.735427086029324, 2.089210378881618, 3.107432433370288, 3.378164263919682, 3.885906800361428, 4.646742716817499, 5.181763559119572, 5.611958857013744, 6.453917546091500, 6.579375311499103, 7.056615650136997, 7.815477564096265, 7.976809474033402, 8.643964519871665, 9.141776769368535, 9.426860444505082, 9.977687847419061, 10.18712179815602, 11.07156763947735, 11.40004703232213, 11.63837354465346, 12.03944738494451, 12.76157833472253
