L(s) = 1 | + 2·2-s + 2·3-s + 4·4-s + 12·5-s + 4·6-s + 8·8-s − 23·9-s + 24·10-s + 48·11-s + 8·12-s − 56·13-s + 24·15-s + 16·16-s + 114·17-s − 46·18-s − 2·19-s + 48·20-s + 96·22-s − 120·23-s + 16·24-s + 19·25-s − 112·26-s − 100·27-s − 54·29-s + 48·30-s − 236·31-s + 32·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.384·3-s + 1/2·4-s + 1.07·5-s + 0.272·6-s + 0.353·8-s − 0.851·9-s + 0.758·10-s + 1.31·11-s + 0.192·12-s − 1.19·13-s + 0.413·15-s + 1/4·16-s + 1.62·17-s − 0.602·18-s − 0.0241·19-s + 0.536·20-s + 0.930·22-s − 1.08·23-s + 0.136·24-s + 0.151·25-s − 0.844·26-s − 0.712·27-s − 0.345·29-s + 0.292·30-s − 1.36·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.902371892\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.902371892\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 56 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 236 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 + 376 T + p^{3} T^{2} \) |
| 47 | \( 1 - 12 T + p^{3} T^{2} \) |
| 53 | \( 1 - 174 T + p^{3} T^{2} \) |
| 59 | \( 1 + 138 T + p^{3} T^{2} \) |
| 61 | \( 1 + 380 T + p^{3} T^{2} \) |
| 67 | \( 1 + 484 T + p^{3} T^{2} \) |
| 71 | \( 1 - 576 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1150 T + p^{3} T^{2} \) |
| 79 | \( 1 - 776 T + p^{3} T^{2} \) |
| 83 | \( 1 + 378 T + p^{3} T^{2} \) |
| 89 | \( 1 - 390 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1330 T + p^{3} T^{2} \) |
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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
−13.73397897870439385301574992926, −12.41452635576454198119214831976, −11.61879940966370156439632035826, −10.06447456248675177451945348105, −9.252572613032144419263243224512, −7.71524436453679388439034948965, −6.25676915027756499817112600586, −5.30769600119045476453389118174, −3.51604532587599459406552309244, −1.98336991228660146622330664729,
1.98336991228660146622330664729, 3.51604532587599459406552309244, 5.30769600119045476453389118174, 6.25676915027756499817112600586, 7.71524436453679388439034948965, 9.252572613032144419263243224512, 10.06447456248675177451945348105, 11.61879940966370156439632035826, 12.41452635576454198119214831976, 13.73397897870439385301574992926