| L(s) = 1 | + 2-s + 4-s + 0.618·5-s + 7-s + 8-s + 0.618·10-s + 1.61·13-s + 14-s + 16-s − 6.23·17-s − 5.85·19-s + 0.618·20-s − 5.09·23-s − 4.61·25-s + 1.61·26-s + 28-s + 4.09·29-s − 2.14·31-s + 32-s − 6.23·34-s + 0.618·35-s − 9·37-s − 5.85·38-s + 0.618·40-s − 2.61·41-s − 4.85·43-s − 5.09·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.276·5-s + 0.377·7-s + 0.353·8-s + 0.195·10-s + 0.448·13-s + 0.267·14-s + 0.250·16-s − 1.51·17-s − 1.34·19-s + 0.138·20-s − 1.06·23-s − 0.923·25-s + 0.317·26-s + 0.188·28-s + 0.759·29-s − 0.385·31-s + 0.176·32-s − 1.06·34-s + 0.104·35-s − 1.47·37-s − 0.949·38-s + 0.0977·40-s − 0.408·41-s − 0.740·43-s − 0.750·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 0.618T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 + 6.23T + 17T^{2} \) |
| 19 | \( 1 + 5.85T + 19T^{2} \) |
| 23 | \( 1 + 5.09T + 23T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 + 9T + 37T^{2} \) |
| 41 | \( 1 + 2.61T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 - 3.61T + 47T^{2} \) |
| 53 | \( 1 - 0.381T + 53T^{2} \) |
| 59 | \( 1 + 5.23T + 59T^{2} \) |
| 61 | \( 1 - 8.32T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 1.47T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 3T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 6.70T + 97T^{2} \) |
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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
−7.56918222652858798439964731574, −6.68922115073590551753767657170, −6.27293286281509943708246731955, −5.53784740138617772289790999648, −4.63598817968049691678443513209, −4.16977126531561948388429199031, −3.30792035003547714650073111638, −2.17011102730932999761942903810, −1.74673698149765650500557782132, 0,
1.74673698149765650500557782132, 2.17011102730932999761942903810, 3.30792035003547714650073111638, 4.16977126531561948388429199031, 4.63598817968049691678443513209, 5.53784740138617772289790999648, 6.27293286281509943708246731955, 6.68922115073590551753767657170, 7.56918222652858798439964731574
