L(s) = 1 | − 2-s + 4-s − 2·5-s + 2.56·7-s − 8-s + 2·10-s − 2.56·11-s − 5.68·13-s − 2.56·14-s + 16-s + 0.561·17-s + 0.561·19-s − 2·20-s + 2.56·22-s + 3.68·23-s − 25-s + 5.68·26-s + 2.56·28-s − 7.12·29-s − 0.876·31-s − 32-s − 0.561·34-s − 5.12·35-s − 37-s − 0.561·38-s + 2·40-s − 2.87·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.894·5-s + 0.968·7-s − 0.353·8-s + 0.632·10-s − 0.772·11-s − 1.57·13-s − 0.684·14-s + 0.250·16-s + 0.136·17-s + 0.128·19-s − 0.447·20-s + 0.546·22-s + 0.768·23-s − 0.200·25-s + 1.11·26-s + 0.484·28-s − 1.32·29-s − 0.157·31-s − 0.176·32-s − 0.0963·34-s − 0.865·35-s − 0.164·37-s − 0.0910·38-s + 0.316·40-s − 0.449·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + 5.68T + 13T^{2} \) |
| 17 | \( 1 - 0.561T + 17T^{2} \) |
| 19 | \( 1 - 0.561T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 + 0.876T + 31T^{2} \) |
| 41 | \( 1 + 2.87T + 41T^{2} \) |
| 43 | \( 1 + 7.12T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 2.56T + 53T^{2} \) |
| 59 | \( 1 + 9.12T + 59T^{2} \) |
| 61 | \( 1 - 4.24T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 0.561T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−10.06914809806822753310533686714, −9.217930316328139086163603069016, −8.095753005299843456753776851171, −7.70942579146276752016297586037, −6.94488942436586737620818264181, −5.39129730831884065767195333794, −4.63626967180425673112389828636, −3.20315597968137169051876135361, −1.89857937519663082123922664115, 0,
1.89857937519663082123922664115, 3.20315597968137169051876135361, 4.63626967180425673112389828636, 5.39129730831884065767195333794, 6.94488942436586737620818264181, 7.70942579146276752016297586037, 8.095753005299843456753776851171, 9.217930316328139086163603069016, 10.06914809806822753310533686714