| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 4·7-s − 8-s + 9-s − 5·11-s + 12-s − 4·14-s + 16-s + 2·17-s − 18-s − 8·19-s + 4·21-s + 5·22-s − 3·23-s − 24-s + 27-s + 4·28-s + 7·29-s + 7·31-s − 32-s − 5·33-s − 2·34-s + 36-s − 3·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s + 0.288·12-s − 1.06·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.83·19-s + 0.872·21-s + 1.06·22-s − 0.625·23-s − 0.204·24-s + 0.192·27-s + 0.755·28-s + 1.29·29-s + 1.25·31-s − 0.176·32-s − 0.870·33-s − 0.342·34-s + 1/6·36-s − 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−15.56616633990507, −15.20529677733671, −14.50544488111628, −14.21439669572087, −13.56003534463482, −12.86798126985815, −12.41742514101583, −11.73234918472849, −11.14670278948996, −10.60535636609719, −10.19591458968971, −9.740330902544243, −8.685465191336352, −8.345384508906511, −8.101635568250355, −7.587441469493495, −6.794257194873883, −6.156545304568024, −5.328535330541719, −4.715072808967547, −4.235239334193185, −3.149284558952560, −2.454486954346960, −1.965036977961395, −1.151219441008331, 0,
1.151219441008331, 1.965036977961395, 2.454486954346960, 3.149284558952560, 4.235239334193185, 4.715072808967547, 5.328535330541719, 6.156545304568024, 6.794257194873883, 7.587441469493495, 8.101635568250355, 8.345384508906511, 8.685465191336352, 9.740330902544243, 10.19591458968971, 10.60535636609719, 11.14670278948996, 11.73234918472849, 12.41742514101583, 12.86798126985815, 13.56003534463482, 14.21439669572087, 14.50544488111628, 15.20529677733671, 15.56616633990507
