L(s) = 1 | − 2.15·2-s + 2.37·3-s + 2.63·4-s − 5-s − 5.11·6-s − 1.36·8-s + 2.64·9-s + 2.15·10-s − 11-s + 6.25·12-s − 5.45·13-s − 2.37·15-s − 2.33·16-s − 0.865·17-s − 5.69·18-s + 2.98·19-s − 2.63·20-s + 2.15·22-s + 6.14·23-s − 3.23·24-s + 25-s + 11.7·26-s − 0.837·27-s − 0.799·29-s + 5.11·30-s − 1.38·31-s + 7.74·32-s + ⋯ |
L(s) = 1 | − 1.52·2-s + 1.37·3-s + 1.31·4-s − 0.447·5-s − 2.08·6-s − 0.481·8-s + 0.882·9-s + 0.680·10-s − 0.301·11-s + 1.80·12-s − 1.51·13-s − 0.613·15-s − 0.584·16-s − 0.209·17-s − 1.34·18-s + 0.685·19-s − 0.588·20-s + 0.458·22-s + 1.28·23-s − 0.659·24-s + 0.200·25-s + 2.30·26-s − 0.161·27-s − 0.148·29-s + 0.933·30-s − 0.249·31-s + 1.36·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 3 | \( 1 - 2.37T + 3T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 + 0.865T + 17T^{2} \) |
| 19 | \( 1 - 2.98T + 19T^{2} \) |
| 23 | \( 1 - 6.14T + 23T^{2} \) |
| 29 | \( 1 + 0.799T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 - 7.02T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 1.49T + 43T^{2} \) |
| 47 | \( 1 + 3.54T + 47T^{2} \) |
| 53 | \( 1 + 8.30T + 53T^{2} \) |
| 59 | \( 1 - 5.74T + 59T^{2} \) |
| 61 | \( 1 - 3.97T + 61T^{2} \) |
| 67 | \( 1 - 1.46T + 67T^{2} \) |
| 71 | \( 1 + 1.38T + 71T^{2} \) |
| 73 | \( 1 + 1.27T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 - 2.46T + 97T^{2} \) |
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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
−8.448562692624902645510478758710, −7.966100310475824317491032214628, −7.27251947665458201101903536126, −6.87513386151300175931244600979, −5.27463115404193944422899031907, −4.36640168315260478110140593162, −3.11760342843965074779477869619, −2.54179993955079052857380990725, −1.48398293947217292054912099066, 0,
1.48398293947217292054912099066, 2.54179993955079052857380990725, 3.11760342843965074779477869619, 4.36640168315260478110140593162, 5.27463115404193944422899031907, 6.87513386151300175931244600979, 7.27251947665458201101903536126, 7.966100310475824317491032214628, 8.448562692624902645510478758710