| L(s) = 1 | − 2-s + 2·3-s − 4-s + 5-s − 2·6-s + 3·8-s + 9-s − 10-s − 2·12-s + 2·13-s + 2·15-s − 16-s + 2·17-s − 18-s − 4·19-s − 20-s + 6·23-s + 6·24-s + 25-s − 2·26-s − 4·27-s − 4·29-s − 2·30-s − 4·31-s − 5·32-s − 2·34-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.447·5-s − 0.816·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.577·12-s + 0.554·13-s + 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 1.25·23-s + 1.22·24-s + 1/5·25-s − 0.392·26-s − 0.769·27-s − 0.742·29-s − 0.365·30-s − 0.718·31-s − 0.883·32-s − 0.342·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−15.35203839512855, −14.68294205478924, −14.39165491026228, −13.91443746031229, −13.31880826776681, −12.89928738814639, −12.57828771517791, −11.36921765698918, −11.09747693070794, −10.30416203944128, −9.934084926061838, −9.246523173333612, −8.841327635355687, −8.612777923621388, −7.880907610427760, −7.438521780435288, −6.761476343961777, −5.956043594123012, −5.262340514958605, −4.675225168266157, −3.729971366620035, −3.461060342962471, −2.493405170422030, −1.828365281978573, −1.134525102202380, 0,
1.134525102202380, 1.828365281978573, 2.493405170422030, 3.461060342962471, 3.729971366620035, 4.675225168266157, 5.262340514958605, 5.956043594123012, 6.761476343961777, 7.438521780435288, 7.880907610427760, 8.612777923621388, 8.841327635355687, 9.246523173333612, 9.934084926061838, 10.30416203944128, 11.09747693070794, 11.36921765698918, 12.57828771517791, 12.89928738814639, 13.31880826776681, 13.91443746031229, 14.39165491026228, 14.68294205478924, 15.35203839512855
