| L(s) = 1 | − 2.34·3-s + 5-s − 1.53·7-s + 2.48·9-s − 2.13·11-s − 0.470·13-s − 2.34·15-s + 2.13·17-s − 4.98·19-s + 3.59·21-s − 2.33·23-s + 25-s + 1.19·27-s + 6.12·29-s + 6.34·31-s + 5.00·33-s − 1.53·35-s + 2.64·37-s + 1.10·39-s + 6.14·41-s − 7.94·43-s + 2.48·45-s + 10.8·47-s − 4.64·49-s − 5.01·51-s + 6.95·53-s − 2.13·55-s + ⋯ |
| L(s) = 1 | − 1.35·3-s + 0.447·5-s − 0.579·7-s + 0.829·9-s − 0.644·11-s − 0.130·13-s − 0.604·15-s + 0.518·17-s − 1.14·19-s + 0.784·21-s − 0.486·23-s + 0.200·25-s + 0.230·27-s + 1.13·29-s + 1.13·31-s + 0.871·33-s − 0.259·35-s + 0.434·37-s + 0.176·39-s + 0.959·41-s − 1.21·43-s + 0.370·45-s + 1.58·47-s − 0.663·49-s − 0.701·51-s + 0.954·53-s − 0.288·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + 2.34T + 3T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 + 2.13T + 11T^{2} \) |
| 13 | \( 1 + 0.470T + 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 + 2.33T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 - 2.64T + 37T^{2} \) |
| 41 | \( 1 - 6.14T + 41T^{2} \) |
| 43 | \( 1 + 7.94T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 6.95T + 53T^{2} \) |
| 59 | \( 1 + 9.95T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 1.12T + 67T^{2} \) |
| 71 | \( 1 + 8.62T + 71T^{2} \) |
| 73 | \( 1 + 1.80T + 73T^{2} \) |
| 79 | \( 1 - 5.78T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 3.80T + 97T^{2} \) |
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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
−7.61638364313202111141663773054, −6.73185707957602736825793059318, −6.14150575900175795972614880900, −5.81505011201059488409810686190, −4.86302815150093951320115582230, −4.38375468695967068466940409257, −3.13957535971519636397049582372, −2.32027982568464189371445496290, −1.03686821551739188128088434589, 0,
1.03686821551739188128088434589, 2.32027982568464189371445496290, 3.13957535971519636397049582372, 4.38375468695967068466940409257, 4.86302815150093951320115582230, 5.81505011201059488409810686190, 6.14150575900175795972614880900, 6.73185707957602736825793059318, 7.61638364313202111141663773054
