| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 4·11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 2·17-s − 18-s + 19-s − 20-s + 21-s + 4·22-s − 4·23-s + 24-s + 25-s − 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 − 1.20·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.218·21-s + 0.852·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−14.98381621006065, −14.24990987866421, −13.49022419331829, −13.22959604124563, −12.61941592944475, −11.89368131014054, −11.79335078924690, −10.96234575479436, −10.66886641364014, −10.07267311456698, −9.727924247605266, −9.002591534296143, −8.229820454414828, −8.165139139303893, −7.307280155986797, −6.876213391858927, −6.361611877483411, −5.633627760579162, −5.153224629018167, −4.548806813026445, −3.562721174507250, −3.292905158410722, −2.228022942657294, −1.757545563103476, −0.6140025902457494, 0,
0.6140025902457494, 1.757545563103476, 2.228022942657294, 3.292905158410722, 3.562721174507250, 4.548806813026445, 5.153224629018167, 5.633627760579162, 6.361611877483411, 6.876213391858927, 7.307280155986797, 8.165139139303893, 8.229820454414828, 9.002591534296143, 9.727924247605266, 10.07267311456698, 10.66886641364014, 10.96234575479436, 11.79335078924690, 11.89368131014054, 12.61941592944475, 13.22959604124563, 13.49022419331829, 14.24990987866421, 14.98381621006065
