| L(s) = 1 | + 3.24·3-s + 5.01·5-s − 22.4·7-s − 16.4·9-s + 50.5·11-s + 89.2·13-s + 16.2·15-s − 66.9·17-s − 97.7·19-s − 72.9·21-s − 23·23-s − 99.8·25-s − 141.·27-s − 29.9·29-s − 73.5·31-s + 163.·33-s − 112.·35-s − 1.12·37-s + 289.·39-s + 34.3·41-s + 160.·43-s − 82.7·45-s − 289.·47-s + 162.·49-s − 217.·51-s + 392.·53-s + 253.·55-s + ⋯ |
| L(s) = 1 | + 0.624·3-s + 0.448·5-s − 1.21·7-s − 0.610·9-s + 1.38·11-s + 1.90·13-s + 0.280·15-s − 0.954·17-s − 1.17·19-s − 0.757·21-s − 0.208·23-s − 0.798·25-s − 1.00·27-s − 0.192·29-s − 0.426·31-s + 0.864·33-s − 0.545·35-s − 0.00500·37-s + 1.18·39-s + 0.130·41-s + 0.569·43-s − 0.273·45-s − 0.898·47-s + 0.474·49-s − 0.595·51-s + 1.01·53-s + 0.621·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 3.24T + 27T^{2} \) |
| 5 | \( 1 - 5.01T + 125T^{2} \) |
| 7 | \( 1 + 22.4T + 343T^{2} \) |
| 11 | \( 1 - 50.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 89.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 66.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 97.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 29.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 73.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 1.12T + 5.06e4T^{2} \) |
| 41 | \( 1 - 34.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 160.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 289.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 392.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 174.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 474.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 671.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 481.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 778.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 132.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 808.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 824.T + 9.12e5T^{2} \) |
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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
−8.887195605291232101115516767968, −8.249958545733039998493398311219, −6.87306210111826920794933608549, −6.25756508616341477133847366171, −5.80388754757634618470750470647, −4.03630209144009521237940898566, −3.67598911253509922370088794556, −2.54145687691683855120552076672, −1.49183817081434813619177605511, 0,
1.49183817081434813619177605511, 2.54145687691683855120552076672, 3.67598911253509922370088794556, 4.03630209144009521237940898566, 5.80388754757634618470750470647, 6.25756508616341477133847366171, 6.87306210111826920794933608549, 8.249958545733039998493398311219, 8.887195605291232101115516767968
