| L(s) = 1 | − 2-s + 3-s + 4-s + 3.47·5-s − 6-s + 3.94·7-s − 8-s + 9-s − 3.47·10-s − 3.52·11-s + 12-s − 0.697·13-s − 3.94·14-s + 3.47·15-s + 16-s + 2.38·17-s − 18-s − 19-s + 3.47·20-s + 3.94·21-s + 3.52·22-s + 3.57·23-s − 24-s + 7.05·25-s + 0.697·26-s + 27-s + 3.94·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.55·5-s − 0.408·6-s + 1.49·7-s − 0.353·8-s + 0.333·9-s − 1.09·10-s − 1.06·11-s + 0.288·12-s − 0.193·13-s − 1.05·14-s + 0.896·15-s + 0.250·16-s + 0.577·17-s − 0.235·18-s − 0.229·19-s + 0.776·20-s + 0.860·21-s + 0.751·22-s + 0.744·23-s − 0.204·24-s + 1.41·25-s + 0.136·26-s + 0.192·27-s + 0.745·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.993726919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.993726919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 - 3.47T + 5T^{2} \) |
| 7 | \( 1 - 3.94T + 7T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 + 0.697T + 13T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 23 | \( 1 - 3.57T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 4.98T + 31T^{2} \) |
| 37 | \( 1 + 2.09T + 37T^{2} \) |
| 41 | \( 1 + 6.99T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 3.95T + 47T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 4.17T + 61T^{2} \) |
| 67 | \( 1 - 9.87T + 67T^{2} \) |
| 71 | \( 1 + 8.71T + 71T^{2} \) |
| 73 | \( 1 - 7.14T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 2.51T + 97T^{2} \) |
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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
−8.210048286619323600491376038442, −7.51925490422130849247409177350, −6.89045472126900489945655515835, −5.85012046582228450084871623864, −5.27973192735444144079404439103, −4.70175221923366083794485544346, −3.33521661087222266511531268151, −2.33744000626422324729040813817, −1.95817261479563760572294462432, −1.04137758671867455270223509537,
1.04137758671867455270223509537, 1.95817261479563760572294462432, 2.33744000626422324729040813817, 3.33521661087222266511531268151, 4.70175221923366083794485544346, 5.27973192735444144079404439103, 5.85012046582228450084871623864, 6.89045472126900489945655515835, 7.51925490422130849247409177350, 8.210048286619323600491376038442
