| L(s) = 1 | − 2·2-s + 2·3-s + 4·4-s + 5·5-s − 4·6-s − 12·7-s − 8·8-s − 23·9-s − 10·10-s − 20·11-s + 8·12-s − 4·13-s + 24·14-s + 10·15-s + 16·16-s − 34·17-s + 46·18-s − 19·19-s + 20·20-s − 24·21-s + 40·22-s + 40·23-s − 16·24-s + 25·25-s + 8·26-s − 100·27-s − 48·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.384·3-s + 1/2·4-s + 0.447·5-s − 0.272·6-s − 0.647·7-s − 0.353·8-s − 0.851·9-s − 0.316·10-s − 0.548·11-s + 0.192·12-s − 0.0853·13-s + 0.458·14-s + 0.172·15-s + 1/4·16-s − 0.485·17-s + 0.602·18-s − 0.229·19-s + 0.223·20-s − 0.249·21-s + 0.387·22-s + 0.362·23-s − 0.136·24-s + 1/5·25-s + 0.0603·26-s − 0.712·27-s − 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 + p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 40 T + p^{3} T^{2} \) |
| 29 | \( 1 + 150 T + p^{3} T^{2} \) |
| 31 | \( 1 + 200 T + p^{3} T^{2} \) |
| 37 | \( 1 + 156 T + p^{3} T^{2} \) |
| 41 | \( 1 + 218 T + p^{3} T^{2} \) |
| 43 | \( 1 - 248 T + p^{3} T^{2} \) |
| 47 | \( 1 + 180 T + p^{3} T^{2} \) |
| 53 | \( 1 - 72 T + p^{3} T^{2} \) |
| 59 | \( 1 + 48 T + p^{3} T^{2} \) |
| 61 | \( 1 + 134 T + p^{3} T^{2} \) |
| 67 | \( 1 - 334 T + p^{3} T^{2} \) |
| 71 | \( 1 + 520 T + p^{3} T^{2} \) |
| 73 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 79 | \( 1 - 980 T + p^{3} T^{2} \) |
| 83 | \( 1 + 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 670 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1124 T + p^{3} T^{2} \) |
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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
−11.37074003556315698531882029448, −10.50853855625788557303792480978, −9.432981437652067298370422283572, −8.773708131602161993835172614011, −7.64671361753113495712537595448, −6.48000227442668949120976705817, −5.36523194234257699335348370860, −3.36109750994214809893125246849, −2.14238964388948572972835276313, 0,
2.14238964388948572972835276313, 3.36109750994214809893125246849, 5.36523194234257699335348370860, 6.48000227442668949120976705817, 7.64671361753113495712537595448, 8.773708131602161993835172614011, 9.432981437652067298370422283572, 10.50853855625788557303792480978, 11.37074003556315698531882029448
