| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s + 11-s − 12-s + 13-s − 2·14-s + 16-s − 17-s − 18-s + 4·19-s − 2·21-s − 22-s + 6·23-s + 24-s − 5·25-s − 26-s − 27-s + 2·28-s − 8·29-s + 2·31-s − 32-s − 33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.436·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s + 0.377·28-s − 1.48·29-s + 0.359·31-s − 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.53258422360301, −15.93525309148417, −15.24392051134339, −14.96774119063243, −14.20329646795771, −13.47766009736874, −13.07196492046423, −12.16393245686485, −11.69921357781111, −11.26474054893049, −10.80238989660613, −10.14153788904966, −9.377484650574977, −9.108171036168251, −8.179318011591290, −7.742956628661186, −7.081031270327253, −6.516955998966931, −5.675397611648862, −5.244769853821002, −4.404792047978282, −3.636218196978220, −2.740746396114859, −1.699608028230151, −1.174951689611294, 0,
1.174951689611294, 1.699608028230151, 2.740746396114859, 3.636218196978220, 4.404792047978282, 5.244769853821002, 5.675397611648862, 6.516955998966931, 7.081031270327253, 7.742956628661186, 8.179318011591290, 9.108171036168251, 9.377484650574977, 10.14153788904966, 10.80238989660613, 11.26474054893049, 11.69921357781111, 12.16393245686485, 13.07196492046423, 13.47766009736874, 14.20329646795771, 14.96774119063243, 15.24392051134339, 15.93525309148417, 16.53258422360301
