| L(s) = 1 | + 2.85·3-s + 1.91·5-s − 1.08·7-s + 5.16·9-s + 11-s + 5.31·13-s + 5.47·15-s + 2·17-s − 19-s − 3.09·21-s + 3.07·23-s − 1.32·25-s + 6.17·27-s + 1.29·29-s − 4.99·31-s + 2.85·33-s − 2.07·35-s − 3.24·37-s + 15.1·39-s + 2.99·41-s + 8.87·43-s + 9.88·45-s − 6.32·47-s − 5.82·49-s + 5.71·51-s + 2·53-s + 1.91·55-s + ⋯ |
| L(s) = 1 | + 1.64·3-s + 0.856·5-s − 0.409·7-s + 1.72·9-s + 0.301·11-s + 1.47·13-s + 1.41·15-s + 0.485·17-s − 0.229·19-s − 0.675·21-s + 0.641·23-s − 0.265·25-s + 1.18·27-s + 0.241·29-s − 0.896·31-s + 0.497·33-s − 0.351·35-s − 0.533·37-s + 2.43·39-s + 0.467·41-s + 1.35·43-s + 1.47·45-s − 0.922·47-s − 0.832·49-s + 0.800·51-s + 0.274·53-s + 0.258·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.340103712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.340103712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.85T + 3T^{2} \) |
| 5 | \( 1 - 1.91T + 5T^{2} \) |
| 7 | \( 1 + 1.08T + 7T^{2} \) |
| 13 | \( 1 - 5.31T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 - 3.07T + 23T^{2} \) |
| 29 | \( 1 - 1.29T + 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 - 2.99T + 41T^{2} \) |
| 43 | \( 1 - 8.87T + 43T^{2} \) |
| 47 | \( 1 + 6.32T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 3.72T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 + 5.78T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 0.168T + 79T^{2} \) |
| 83 | \( 1 + 1.58T + 83T^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.896331233424734676225030688983, −7.980366216542983275018644690460, −7.33011684687945427328965469376, −6.35415802555324251430529680258, −5.80731614375410913631385333646, −4.56517972860610334833553134219, −3.58939469321543617510404624497, −3.13982413445210366364727712954, −2.08434337102373577935838307825, −1.32003259419381214007087611847,
1.32003259419381214007087611847, 2.08434337102373577935838307825, 3.13982413445210366364727712954, 3.58939469321543617510404624497, 4.56517972860610334833553134219, 5.80731614375410913631385333646, 6.35415802555324251430529680258, 7.33011684687945427328965469376, 7.980366216542983275018644690460, 8.896331233424734676225030688983
