| L(s) = 1 | − 0.484·2-s − 1.76·4-s − 2.26·5-s + 1.63·7-s + 1.82·8-s + 1.09·10-s − 0.792·11-s + 0.637·13-s − 0.792·14-s + 2.64·16-s + 0.222·17-s + 7.75·19-s + 3.99·20-s + 0.383·22-s − 6.98·23-s + 0.128·25-s − 0.308·26-s − 2.89·28-s − 3.66·29-s − 1.51·31-s − 4.92·32-s − 0.107·34-s − 3.70·35-s − 2.52·37-s − 3.75·38-s − 4.12·40-s + 3.38·41-s + ⋯ |
| L(s) = 1 | − 0.342·2-s − 0.882·4-s − 1.01·5-s + 0.618·7-s + 0.644·8-s + 0.346·10-s − 0.239·11-s + 0.176·13-s − 0.211·14-s + 0.662·16-s + 0.0538·17-s + 1.77·19-s + 0.894·20-s + 0.0818·22-s − 1.45·23-s + 0.0256·25-s − 0.0605·26-s − 0.546·28-s − 0.680·29-s − 0.271·31-s − 0.871·32-s − 0.0184·34-s − 0.626·35-s − 0.414·37-s − 0.608·38-s − 0.652·40-s + 0.528·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 0.484T + 2T^{2} \) |
| 5 | \( 1 + 2.26T + 5T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 + 0.792T + 11T^{2} \) |
| 13 | \( 1 - 0.637T + 13T^{2} \) |
| 17 | \( 1 - 0.222T + 17T^{2} \) |
| 19 | \( 1 - 7.75T + 19T^{2} \) |
| 23 | \( 1 + 6.98T + 23T^{2} \) |
| 29 | \( 1 + 3.66T + 29T^{2} \) |
| 31 | \( 1 + 1.51T + 31T^{2} \) |
| 37 | \( 1 + 2.52T + 37T^{2} \) |
| 41 | \( 1 - 3.38T + 41T^{2} \) |
| 43 | \( 1 + 4.81T + 43T^{2} \) |
| 47 | \( 1 + 5.25T + 47T^{2} \) |
| 53 | \( 1 - 8.50T + 53T^{2} \) |
| 59 | \( 1 + 6.54T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 2.54T + 71T^{2} \) |
| 73 | \( 1 + 0.150T + 73T^{2} \) |
| 79 | \( 1 - 4.22T + 79T^{2} \) |
| 83 | \( 1 + 3.93T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 5.97T + 97T^{2} \) |
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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
−9.380803254725128618885658500750, −8.459271801273782597584030702200, −7.80530800121271750051398963128, −7.38441916968084545378969095339, −5.83931528728388982847793171298, −4.98106809944589618713419464844, −4.12498171055860450153421262354, −3.31068836199948105588713646681, −1.50377227904679219110838107341, 0,
1.50377227904679219110838107341, 3.31068836199948105588713646681, 4.12498171055860450153421262354, 4.98106809944589618713419464844, 5.83931528728388982847793171298, 7.38441916968084545378969095339, 7.80530800121271750051398963128, 8.459271801273782597584030702200, 9.380803254725128618885658500750
