| L(s) = 1 | + 4.83·2-s + 3·3-s + 15.4·4-s − 21.1·5-s + 14.5·6-s − 16.2·7-s + 35.8·8-s + 9·9-s − 102.·10-s − 30.7·11-s + 46.2·12-s − 78.7·14-s − 63.5·15-s + 49.9·16-s + 46.2·17-s + 43.5·18-s − 144.·19-s − 326.·20-s − 48.8·21-s − 148.·22-s + 8.38·23-s + 107.·24-s + 324.·25-s + 27·27-s − 250.·28-s − 242.·29-s − 307.·30-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.577·3-s + 1.92·4-s − 1.89·5-s + 0.987·6-s − 0.879·7-s + 1.58·8-s + 0.333·9-s − 3.24·10-s − 0.842·11-s + 1.11·12-s − 1.50·14-s − 1.09·15-s + 0.781·16-s + 0.659·17-s + 0.570·18-s − 1.74·19-s − 3.65·20-s − 0.507·21-s − 1.44·22-s + 0.0759·23-s + 0.913·24-s + 2.59·25-s + 0.192·27-s − 1.69·28-s − 1.55·29-s − 1.87·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.83T + 8T^{2} \) |
| 5 | \( 1 + 21.1T + 125T^{2} \) |
| 7 | \( 1 + 16.2T + 343T^{2} \) |
| 11 | \( 1 + 30.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 46.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 8.38T + 1.21e4T^{2} \) |
| 29 | \( 1 + 242.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 87.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 49.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 107.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 35.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 374.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 348.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 679.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 230.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 295.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 329.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 48.9T + 3.89e5T^{2} \) |
| 79 | \( 1 + 107.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 515.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 984.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 487.T + 9.12e5T^{2} \) |
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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
−10.50599892773004810814282894609, −8.982986470902407015326050215181, −7.87759122219341522482651300157, −7.25480218427107124926088438363, −6.24677436555775690706213042287, −4.98487740196664712955712030352, −3.96714050506421761936182903612, −3.51578285362880660613329750206, −2.49482506138021025679583311109, 0,
2.49482506138021025679583311109, 3.51578285362880660613329750206, 3.96714050506421761936182903612, 4.98487740196664712955712030352, 6.24677436555775690706213042287, 7.25480218427107124926088438363, 7.87759122219341522482651300157, 8.982986470902407015326050215181, 10.50599892773004810814282894609
