| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 3·7-s + 8-s − 2·9-s − 10-s + 12-s − 4·13-s + 3·14-s − 15-s + 16-s + 2·17-s − 2·18-s + 5·19-s − 20-s + 3·21-s + 24-s − 4·25-s − 4·26-s − 5·27-s + 3·28-s + 29-s − 30-s − 4·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.13·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.288·12-s − 1.10·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.471·18-s + 1.14·19-s − 0.223·20-s + 0.654·21-s + 0.204·24-s − 4/5·25-s − 0.784·26-s − 0.962·27-s + 0.566·28-s + 0.185·29-s − 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.51351831263502, −14.01522692203184, −13.73330789856100, −13.18683618152990, −12.27059758585230, −11.97636396179033, −11.76385653492319, −11.06680411994014, −10.62797442378249, −9.914502664519249, −9.392049680322110, −8.789809918784204, −8.100441432114314, −7.844235647978834, −7.317294422263816, −6.834054383866913, −5.875676410735176, −5.293113174484750, −5.107212355641982, −4.328281776429541, −3.632954954053875, −3.256033105759690, −2.417590526342406, −1.994756696637845, −1.123636595932889, 0,
1.123636595932889, 1.994756696637845, 2.417590526342406, 3.256033105759690, 3.632954954053875, 4.328281776429541, 5.107212355641982, 5.293113174484750, 5.875676410735176, 6.834054383866913, 7.317294422263816, 7.844235647978834, 8.100441432114314, 8.789809918784204, 9.392049680322110, 9.914502664519249, 10.62797442378249, 11.06680411994014, 11.76385653492319, 11.97636396179033, 12.27059758585230, 13.18683618152990, 13.73330789856100, 14.01522692203184, 14.51351831263502
