| L(s) = 1 | − 1.73·2-s − 1.80·3-s + 1.01·4-s + 5-s + 3.13·6-s − 4.26·7-s + 1.70·8-s + 0.265·9-s − 1.73·10-s − 11-s − 1.83·12-s + 6.42·13-s + 7.41·14-s − 1.80·15-s − 4.99·16-s + 7.02·17-s − 0.460·18-s + 1.48·19-s + 1.01·20-s + 7.71·21-s + 1.73·22-s − 6.33·23-s − 3.08·24-s + 25-s − 11.1·26-s + 4.94·27-s − 4.33·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 1.04·3-s + 0.507·4-s + 0.447·5-s + 1.28·6-s − 1.61·7-s + 0.604·8-s + 0.0884·9-s − 0.549·10-s − 0.301·11-s − 0.529·12-s + 1.78·13-s + 1.98·14-s − 0.466·15-s − 1.24·16-s + 1.70·17-s − 0.108·18-s + 0.340·19-s + 0.227·20-s + 1.68·21-s + 0.370·22-s − 1.32·23-s − 0.630·24-s + 0.200·25-s − 2.18·26-s + 0.950·27-s − 0.819·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5097655492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5097655492\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 + 1.80T + 3T^{2} \) |
| 7 | \( 1 + 4.26T + 7T^{2} \) |
| 13 | \( 1 - 6.42T + 13T^{2} \) |
| 17 | \( 1 - 7.02T + 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 + 6.33T + 23T^{2} \) |
| 29 | \( 1 - 6.42T + 29T^{2} \) |
| 31 | \( 1 - 7.84T + 31T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 - 7.28T + 41T^{2} \) |
| 43 | \( 1 + 8.93T + 43T^{2} \) |
| 47 | \( 1 + 4.65T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 7.63T + 59T^{2} \) |
| 61 | \( 1 + 0.203T + 61T^{2} \) |
| 67 | \( 1 - 6.11T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 79 | \( 1 - 5.80T + 79T^{2} \) |
| 83 | \( 1 - 1.86T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 - 9.86T + 97T^{2} \) |
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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
−8.323564186847000424138216064979, −8.060576596786147670670869953344, −6.79143931574989739684022811483, −6.29280558745267362544144420245, −5.84358561663834351205109693509, −4.90249377626895346030463334933, −3.67999378125086628269818354300, −2.90223247145871715472154453806, −1.37244141146915605955689947886, −0.57805992177934859257835974650,
0.57805992177934859257835974650, 1.37244141146915605955689947886, 2.90223247145871715472154453806, 3.67999378125086628269818354300, 4.90249377626895346030463334933, 5.84358561663834351205109693509, 6.29280558745267362544144420245, 6.79143931574989739684022811483, 8.060576596786147670670869953344, 8.323564186847000424138216064979
