| L(s) = 1 | − 2.70·2-s − 0.701·3-s − 0.701·4-s + 1.89·6-s + 23.5·8-s − 26.5·9-s + 4.01·11-s + 0.492·12-s − 51.6·13-s − 57.8·16-s + 67.5·17-s + 71.6·18-s + 50.9·19-s − 10.8·22-s + 0.507·23-s − 16.4·24-s + 139.·26-s + 37.5·27-s − 120.·29-s + 292.·31-s − 31.6·32-s − 2.81·33-s − 182.·34-s + 18.5·36-s − 144.·37-s − 137.·38-s + 36.2·39-s + ⋯ |
| L(s) = 1 | − 0.955·2-s − 0.135·3-s − 0.0876·4-s + 0.128·6-s + 1.03·8-s − 0.981·9-s + 0.110·11-s + 0.0118·12-s − 1.10·13-s − 0.904·16-s + 0.963·17-s + 0.937·18-s + 0.614·19-s − 0.105·22-s + 0.00460·23-s − 0.140·24-s + 1.05·26-s + 0.267·27-s − 0.768·29-s + 1.69·31-s − 0.174·32-s − 0.0148·33-s − 0.919·34-s + 0.0860·36-s − 0.644·37-s − 0.587·38-s + 0.148·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.70T + 8T^{2} \) |
| 3 | \( 1 + 0.701T + 27T^{2} \) |
| 11 | \( 1 - 4.01T + 1.33e3T^{2} \) |
| 13 | \( 1 + 51.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 50.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.507T + 1.21e4T^{2} \) |
| 29 | \( 1 + 120.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 292.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 144.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 57.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 283.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 233.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 406.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 577.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 322.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 985.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 692.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 428.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 537.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 802.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.75e3T + 9.12e5T^{2} \) |
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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
−8.984108025108986709143319602306, −8.158642560587216430483903533226, −7.60622166253133537069668974346, −6.64056578114174210483766899286, −5.46182483415983989623393484077, −4.82996946225331927094004157420, −3.53882262303398547005675777096, −2.39100152019961134793319251938, −1.04002364931784367478730151222, 0,
1.04002364931784367478730151222, 2.39100152019961134793319251938, 3.53882262303398547005675777096, 4.82996946225331927094004157420, 5.46182483415983989623393484077, 6.64056578114174210483766899286, 7.60622166253133537069668974346, 8.158642560587216430483903533226, 8.984108025108986709143319602306
