| L(s) = 1 | + 3-s + 1.23·7-s + 9-s + 3.23·11-s + 2.61·17-s − 0.854·19-s + 1.23·21-s − 1.85·23-s + 27-s − 0.472·29-s + 5.38·31-s + 3.23·33-s + 2·37-s + 4.47·41-s − 7.70·43-s − 2.09·47-s − 5.47·49-s + 2.61·51-s + 6.09·53-s − 0.854·57-s + 6.76·59-s + 14.5·61-s + 1.23·63-s − 2.29·67-s − 1.85·69-s + 14.4·71-s − 12.4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.467·7-s + 0.333·9-s + 0.975·11-s + 0.634·17-s − 0.195·19-s + 0.269·21-s − 0.386·23-s + 0.192·27-s − 0.0876·29-s + 0.966·31-s + 0.563·33-s + 0.328·37-s + 0.698·41-s − 1.17·43-s − 0.304·47-s − 0.781·49-s + 0.366·51-s + 0.836·53-s − 0.113·57-s + 0.880·59-s + 1.86·61-s + 0.155·63-s − 0.279·67-s − 0.223·69-s + 1.71·71-s − 1.45·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.393564463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.393564463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 2.61T + 17T^{2} \) |
| 19 | \( 1 + 0.854T + 19T^{2} \) |
| 23 | \( 1 + 1.85T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 - 5.38T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 - 6.09T + 53T^{2} \) |
| 59 | \( 1 - 6.76T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 7.14T + 79T^{2} \) |
| 83 | \( 1 - 9.32T + 83T^{2} \) |
| 89 | \( 1 + 7.23T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−9.545955205324220209747433534702, −8.528901317753171730234507680772, −8.091825716177055012271436634771, −7.09468355606880206094451188679, −6.34021739540378920296657153692, −5.28297436217635435043479581423, −4.28671484700009877843806833572, −3.50297794264268562065097464863, −2.32337549506410361358651943355, −1.18225617720762222381236234885,
1.18225617720762222381236234885, 2.32337549506410361358651943355, 3.50297794264268562065097464863, 4.28671484700009877843806833572, 5.28297436217635435043479581423, 6.34021739540378920296657153692, 7.09468355606880206094451188679, 8.091825716177055012271436634771, 8.528901317753171730234507680772, 9.545955205324220209747433534702
