L(s) = 1 | − 5-s − 1.42·7-s + 2.67·11-s + 4.67·13-s − 2.67·17-s − 4.67·19-s − 5.91·23-s + 25-s + 9.48·29-s − 6.96·31-s + 1.42·35-s − 1.81·37-s + 1.47·41-s − 0.471·43-s + 6.95·47-s − 4.96·49-s + 1.14·53-s − 2.67·55-s − 1.14·59-s − 2.52·61-s − 4.67·65-s + 6.59·67-s − 12.8·71-s − 1.71·73-s − 3.81·77-s + 0.287·79-s − 4.28·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.539·7-s + 0.805·11-s + 1.29·13-s − 0.648·17-s − 1.07·19-s − 1.23·23-s + 0.200·25-s + 1.76·29-s − 1.25·31-s + 0.241·35-s − 0.298·37-s + 0.229·41-s − 0.0718·43-s + 1.01·47-s − 0.708·49-s + 0.157·53-s − 0.360·55-s − 0.148·59-s − 0.323·61-s − 0.579·65-s + 0.805·67-s − 1.52·71-s − 0.200·73-s − 0.434·77-s + 0.0323·79-s − 0.470·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 + 1.42T + 7T^{2} \) |
| 11 | \( 1 - 2.67T + 11T^{2} \) |
| 13 | \( 1 - 4.67T + 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 + 5.91T + 23T^{2} \) |
| 29 | \( 1 - 9.48T + 29T^{2} \) |
| 31 | \( 1 + 6.96T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 + 0.471T + 43T^{2} \) |
| 47 | \( 1 - 6.95T + 47T^{2} \) |
| 53 | \( 1 - 1.14T + 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 - 6.59T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 1.71T + 73T^{2} \) |
| 79 | \( 1 - 0.287T + 79T^{2} \) |
| 83 | \( 1 + 4.28T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 7.83T + 97T^{2} \) |
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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
−7.70863641178134995948279617821, −6.75734558522214861351480366884, −6.37284268330100743630471394850, −5.73583001602024563848570320783, −4.53932889703234892069551018581, −3.99780009097491164780791302249, −3.36904144052067335284846894371, −2.29288263752100290951924085817, −1.26967826730000423390993307481, 0,
1.26967826730000423390993307481, 2.29288263752100290951924085817, 3.36904144052067335284846894371, 3.99780009097491164780791302249, 4.53932889703234892069551018581, 5.73583001602024563848570320783, 6.37284268330100743630471394850, 6.75734558522214861351480366884, 7.70863641178134995948279617821