| L(s) = 1 | + 2.12·2-s − 3.21·3-s + 2.49·4-s + 1.10·5-s − 6.81·6-s − 3.09·7-s + 1.05·8-s + 7.31·9-s + 2.35·10-s + 3.71·11-s − 8.01·12-s − 6.57·14-s − 3.56·15-s − 2.76·16-s + 2.35·17-s + 15.5·18-s + 8.07·19-s + 2.76·20-s + 9.95·21-s + 7.87·22-s − 6.81·23-s − 3.37·24-s − 3.76·25-s − 13.8·27-s − 7.73·28-s + 2.35·29-s − 7.55·30-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 1.85·3-s + 1.24·4-s + 0.496·5-s − 2.78·6-s − 1.17·7-s + 0.371·8-s + 2.43·9-s + 0.743·10-s + 1.11·11-s − 2.31·12-s − 1.75·14-s − 0.920·15-s − 0.690·16-s + 0.571·17-s + 3.65·18-s + 1.85·19-s + 0.619·20-s + 2.17·21-s + 1.67·22-s − 1.42·23-s − 0.689·24-s − 0.753·25-s − 2.66·27-s − 1.46·28-s + 0.437·29-s − 1.37·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.274312731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.274312731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.12T + 2T^{2} \) |
| 3 | \( 1 + 3.21T + 3T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 7 | \( 1 + 3.09T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 - 8.07T + 19T^{2} \) |
| 23 | \( 1 + 6.81T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 37 | \( 1 - 5.91T + 37T^{2} \) |
| 41 | \( 1 + 4.45T + 41T^{2} \) |
| 43 | \( 1 + 7.80T + 43T^{2} \) |
| 47 | \( 1 + 4.86T + 47T^{2} \) |
| 53 | \( 1 - 3.61T + 53T^{2} \) |
| 59 | \( 1 - 3.22T + 59T^{2} \) |
| 61 | \( 1 - 1.59T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 2.18T + 83T^{2} \) |
| 89 | \( 1 - 6.95T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.76564423369441431363102522363, −6.79885965196830434172357806417, −6.45212555071102938126138287050, −5.92302242215647996063785835615, −5.40984224124731108921861873545, −4.74151015829929253466161792610, −3.82272956641613483241021805458, −3.34475298261391561134882689890, −1.91017322361007630775701548840, −0.71834295126473074901185190565,
0.71834295126473074901185190565, 1.91017322361007630775701548840, 3.34475298261391561134882689890, 3.82272956641613483241021805458, 4.74151015829929253466161792610, 5.40984224124731108921861873545, 5.92302242215647996063785835615, 6.45212555071102938126138287050, 6.79885965196830434172357806417, 7.76564423369441431363102522363
