| L(s) = 1 | − 2-s + 3.34·3-s + 4-s + 2.34·5-s − 3.34·6-s + 4.77·7-s − 8-s + 8.19·9-s − 2.34·10-s + 3.34·12-s − 1.57·13-s − 4.77·14-s + 7.85·15-s + 16-s − 3.77·17-s − 8.19·18-s − 19-s + 2.34·20-s + 15.9·21-s − 2.57·23-s − 3.34·24-s + 0.505·25-s + 1.57·26-s + 17.3·27-s + 4.77·28-s + 4.85·29-s − 7.85·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.93·3-s + 0.5·4-s + 1.04·5-s − 1.36·6-s + 1.80·7-s − 0.353·8-s + 2.73·9-s − 0.741·10-s + 0.965·12-s − 0.437·13-s − 1.27·14-s + 2.02·15-s + 0.250·16-s − 0.915·17-s − 1.93·18-s − 0.229·19-s + 0.524·20-s + 3.48·21-s − 0.537·23-s − 0.683·24-s + 0.101·25-s + 0.309·26-s + 3.34·27-s + 0.902·28-s + 0.900·29-s − 1.43·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.465952007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.465952007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.34T + 3T^{2} \) |
| 5 | \( 1 - 2.34T + 5T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 + 3.77T + 17T^{2} \) |
| 23 | \( 1 + 2.57T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 - 6.69T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 7.00T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 + 7.19T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 - 4.92T + 59T^{2} \) |
| 61 | \( 1 - 4.04T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 2.31T + 71T^{2} \) |
| 73 | \( 1 - 7.77T + 73T^{2} \) |
| 79 | \( 1 + 0.992T + 79T^{2} \) |
| 83 | \( 1 - 8.34T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.410293792625343962730689060979, −8.052506392252050819324583909344, −7.06735772465048582639621691858, −6.56633868167507943826104119567, −5.13916584474663823300265323969, −4.60260322193221114252244589636, −3.53581338379209747134411225842, −2.44280148020333503669290156801, −1.97280050037077129562492170259, −1.42486077276403594445916255565,
1.42486077276403594445916255565, 1.97280050037077129562492170259, 2.44280148020333503669290156801, 3.53581338379209747134411225842, 4.60260322193221114252244589636, 5.13916584474663823300265323969, 6.56633868167507943826104119567, 7.06735772465048582639621691858, 8.052506392252050819324583909344, 8.410293792625343962730689060979
