| L(s) = 1 | + 1.78·2-s − 1.50·3-s + 1.18·4-s − 2.69·6-s − 1.32·7-s − 1.45·8-s − 0.724·9-s − 11-s − 1.78·12-s + 2.54·13-s − 2.36·14-s − 4.96·16-s + 17-s − 1.29·18-s − 5.38·19-s + 2·21-s − 1.78·22-s − 2.64·23-s + 2.19·24-s + 4.53·26-s + 5.61·27-s − 1.56·28-s + 0.584·29-s + 0.916·31-s − 5.94·32-s + 1.50·33-s + 1.78·34-s + ⋯ |
| L(s) = 1 | + 1.26·2-s − 0.870·3-s + 0.591·4-s − 1.09·6-s − 0.501·7-s − 0.515·8-s − 0.241·9-s − 0.301·11-s − 0.514·12-s + 0.704·13-s − 0.632·14-s − 1.24·16-s + 0.242·17-s − 0.304·18-s − 1.23·19-s + 0.436·21-s − 0.380·22-s − 0.550·23-s + 0.448·24-s + 0.889·26-s + 1.08·27-s − 0.296·28-s + 0.108·29-s + 0.164·31-s − 1.05·32-s + 0.262·33-s + 0.305·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.668907440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.668907440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 1.78T + 2T^{2} \) |
| 3 | \( 1 + 1.50T + 3T^{2} \) |
| 7 | \( 1 + 1.32T + 7T^{2} \) |
| 13 | \( 1 - 2.54T + 13T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 - 0.584T + 29T^{2} \) |
| 31 | \( 1 - 0.916T + 31T^{2} \) |
| 37 | \( 1 - 8.04T + 37T^{2} \) |
| 41 | \( 1 - 2.54T + 41T^{2} \) |
| 43 | \( 1 + 1.32T + 43T^{2} \) |
| 47 | \( 1 - 5.03T + 47T^{2} \) |
| 53 | \( 1 - 5.95T + 53T^{2} \) |
| 59 | \( 1 - 6.90T + 59T^{2} \) |
| 61 | \( 1 - 3.20T + 61T^{2} \) |
| 67 | \( 1 - 9.91T + 67T^{2} \) |
| 71 | \( 1 + 9.13T + 71T^{2} \) |
| 73 | \( 1 + 1.91T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 - 19.6T + 97T^{2} \) |
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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
−8.376932118853216345711914098025, −7.26589161341369199360278824156, −6.28107975508800047382231358093, −6.12407417512236884685829026776, −5.42534021880149405088602261404, −4.60426164329440811003056724147, −3.97105051041924211335749107578, −3.10884724330932900568878098744, −2.25841631077319933590654974365, −0.59125606293068858768890167400,
0.59125606293068858768890167400, 2.25841631077319933590654974365, 3.10884724330932900568878098744, 3.97105051041924211335749107578, 4.60426164329440811003056724147, 5.42534021880149405088602261404, 6.12407417512236884685829026776, 6.28107975508800047382231358093, 7.26589161341369199360278824156, 8.376932118853216345711914098025
