L(s) = 1 | + 1.73·2-s − 2.73·3-s + 0.999·4-s − 4.73·6-s + 2.73·7-s − 1.73·8-s + 4.46·9-s + 4.73·11-s − 2.73·12-s + 4·13-s + 4.73·14-s − 5·16-s + 17-s + 7.73·18-s − 1.46·19-s − 7.46·21-s + 8.19·22-s + 8.19·23-s + 4.73·24-s + 6.92·26-s − 3.99·27-s + 2.73·28-s − 3.46·29-s + 3.26·31-s − 5.19·32-s − 12.9·33-s + 1.73·34-s + ⋯ |
L(s) = 1 | + 1.22·2-s − 1.57·3-s + 0.499·4-s − 1.93·6-s + 1.03·7-s − 0.612·8-s + 1.48·9-s + 1.42·11-s − 0.788·12-s + 1.10·13-s + 1.26·14-s − 1.25·16-s + 0.242·17-s + 1.82·18-s − 0.335·19-s − 1.62·21-s + 1.74·22-s + 1.70·23-s + 0.965·24-s + 1.35·26-s − 0.769·27-s + 0.516·28-s − 0.643·29-s + 0.586·31-s − 0.918·32-s − 2.25·33-s + 0.297·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.738260239\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.738260239\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 3.26T + 31T^{2} \) |
| 37 | \( 1 - 0.535T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 0.535T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 2.53T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 6.39T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 + 8.53T + 83T^{2} \) |
| 89 | \( 1 - 4.39T + 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
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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
−11.24678025274335182703938391968, −11.04296065343636595689534756771, −9.487877910666509588169201969638, −8.414091983373977338876427447773, −6.78996604295922861728310179721, −6.25265760195873851031128989022, −5.25317764825741969105188167366, −4.62364220456563052141525776813, −3.59474517355167782177825160326, −1.30545229801566080766941847383,
1.30545229801566080766941847383, 3.59474517355167782177825160326, 4.62364220456563052141525776813, 5.25317764825741969105188167366, 6.25265760195873851031128989022, 6.78996604295922861728310179721, 8.414091983373977338876427447773, 9.487877910666509588169201969638, 11.04296065343636595689534756771, 11.24678025274335182703938391968