| L(s) = 1 | + 2-s + 4-s + 0.414·5-s + 8-s + 0.414·10-s + 11-s + 1.17·13-s + 16-s + 2.17·17-s − 0.828·19-s + 0.414·20-s + 22-s − 3.24·23-s − 4.82·25-s + 1.17·26-s + 2.82·29-s + 6.48·31-s + 32-s + 2.17·34-s + 9.65·37-s − 0.828·38-s + 0.414·40-s − 4.65·41-s − 2.82·43-s + 44-s − 3.24·46-s + 9.24·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.185·5-s + 0.353·8-s + 0.130·10-s + 0.301·11-s + 0.324·13-s + 0.250·16-s + 0.526·17-s − 0.190·19-s + 0.0926·20-s + 0.213·22-s − 0.676·23-s − 0.965·25-s + 0.229·26-s + 0.525·29-s + 1.16·31-s + 0.176·32-s + 0.372·34-s + 1.58·37-s − 0.134·38-s + 0.0654·40-s − 0.727·41-s − 0.431·43-s + 0.150·44-s − 0.478·46-s + 1.34·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.791015634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.791015634\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 0.414T + 5T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 - 9.65T + 37T^{2} \) |
| 41 | \( 1 + 4.65T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 - 9.24T + 47T^{2} \) |
| 53 | \( 1 + 5.17T + 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 - 1.58T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 - 4.75T + 79T^{2} \) |
| 83 | \( 1 - 9.82T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.74508030871712014958543519534, −6.77625810119612415077633585751, −6.21059010224548089020215803050, −5.72383782792806327150762259784, −4.86021416685617960625481715593, −4.19459137599238394008301547240, −3.53573991178784864703653255170, −2.67916067054928753832559166848, −1.87934919272965384754664742806, −0.847669397327943641882464577860,
0.847669397327943641882464577860, 1.87934919272965384754664742806, 2.67916067054928753832559166848, 3.53573991178784864703653255170, 4.19459137599238394008301547240, 4.86021416685617960625481715593, 5.72383782792806327150762259784, 6.21059010224548089020215803050, 6.77625810119612415077633585751, 7.74508030871712014958543519534
