| L(s) = 1 | − 1.41·2-s + 5-s − 3.41·7-s + 2.82·8-s − 1.41·10-s + 11-s + 2.24·13-s + 4.82·14-s − 4.00·16-s − 3.41·17-s − 19-s − 1.41·22-s + 3·23-s − 4·25-s − 3.17·26-s + 6.24·29-s − 6.41·31-s + 4.82·34-s − 3.41·35-s + 10.0·37-s + 1.41·38-s + 2.82·40-s + 1.65·41-s + 0.343·43-s + ⋯ |
| L(s) = 1 | − 1.00·2-s + 0.447·5-s − 1.29·7-s + 0.999·8-s − 0.447·10-s + 0.301·11-s + 0.621·13-s + 1.29·14-s − 1.00·16-s − 0.828·17-s − 0.229·19-s − 0.301·22-s + 0.625·23-s − 0.800·25-s − 0.621·26-s + 1.15·29-s − 1.15·31-s + 0.828·34-s − 0.577·35-s + 1.65·37-s + 0.229·38-s + 0.447·40-s + 0.258·41-s + 0.0523·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 + 3.41T + 7T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 + 3.41T + 17T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 - 6.24T + 29T^{2} \) |
| 31 | \( 1 + 6.41T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 1.65T + 41T^{2} \) |
| 43 | \( 1 - 0.343T + 43T^{2} \) |
| 47 | \( 1 + 8.82T + 47T^{2} \) |
| 53 | \( 1 - 4.48T + 53T^{2} \) |
| 59 | \( 1 - 1.58T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4.48T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 3.41T + 83T^{2} \) |
| 89 | \( 1 + 4.89T + 89T^{2} \) |
| 97 | \( 1 - 2.41T + 97T^{2} \) |
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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
−9.029616056575936623640895012181, −8.288545594411779593161493154789, −7.32557306461599310872818866419, −6.53131322980631352966087170390, −5.91388422041459863675062062540, −4.65982749344289371084893169663, −3.77064372604746606096012529394, −2.61645048657172399571406500208, −1.34977452635185729534176938195, 0,
1.34977452635185729534176938195, 2.61645048657172399571406500208, 3.77064372604746606096012529394, 4.65982749344289371084893169663, 5.91388422041459863675062062540, 6.53131322980631352966087170390, 7.32557306461599310872818866419, 8.288545594411779593161493154789, 9.029616056575936623640895012181
