| L(s) = 1 | − 2-s − 2·3-s + 4-s − 4·5-s + 2·6-s − 2·7-s − 8-s + 9-s + 4·10-s − 4·11-s − 2·12-s − 5·13-s + 2·14-s + 8·15-s + 16-s − 3·17-s − 18-s − 7·19-s − 4·20-s + 4·21-s + 4·22-s − 7·23-s + 2·24-s + 11·25-s + 5·26-s + 4·27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 1.20·11-s − 0.577·12-s − 1.38·13-s + 0.534·14-s + 2.06·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 1.60·19-s − 0.894·20-s + 0.872·21-s + 0.852·22-s − 1.45·23-s + 0.408·24-s + 11/5·25-s + 0.980·26-s + 0.769·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 12323 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−16.31272071088481, −15.72143464045792, −15.47540734360532, −14.77861936729022, −14.34606626067233, −13.00798810808995, −12.77197640709303, −12.38444124436586, −11.59239293558500, −11.44806456384018, −10.80735706479469, −10.24892303718185, −9.922540885161621, −8.849510499795015, −8.502173686713418, −7.658478133873037, −7.481119077275499, −6.730038770287281, −6.215670003138912, −5.485317920271846, −4.683896273351854, −4.302564550661342, −3.366128582687936, −2.696442876181928, −1.811913083803826, 0, 0, 0,
1.811913083803826, 2.696442876181928, 3.366128582687936, 4.302564550661342, 4.683896273351854, 5.485317920271846, 6.215670003138912, 6.730038770287281, 7.481119077275499, 7.658478133873037, 8.502173686713418, 8.849510499795015, 9.922540885161621, 10.24892303718185, 10.80735706479469, 11.44806456384018, 11.59239293558500, 12.38444124436586, 12.77197640709303, 13.00798810808995, 14.34606626067233, 14.77861936729022, 15.47540734360532, 15.72143464045792, 16.31272071088481
