L(s) = 1 | + 2-s + 2·3-s + 4-s + 2.73·5-s + 2·6-s + 7-s + 8-s + 9-s + 2.73·10-s − 2.73·11-s + 2·12-s − 5.46·13-s + 14-s + 5.46·15-s + 16-s + 18-s − 3.46·19-s + 2.73·20-s + 2·21-s − 2.73·22-s − 2·23-s + 2·24-s + 2.46·25-s − 5.46·26-s − 4·27-s + 28-s − 7.66·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 0.5·4-s + 1.22·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.863·10-s − 0.823·11-s + 0.577·12-s − 1.51·13-s + 0.267·14-s + 1.41·15-s + 0.250·16-s + 0.235·18-s − 0.794·19-s + 0.610·20-s + 0.436·21-s − 0.582·22-s − 0.417·23-s + 0.408·24-s + 0.492·25-s − 1.07·26-s − 0.769·27-s + 0.188·28-s − 1.42·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.460168420\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.460168420\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 - 0.535T + 47T^{2} \) |
| 53 | \( 1 - 7.66T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 - 6.73T + 61T^{2} \) |
| 67 | \( 1 - 0.196T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 8.92T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.57153349817273593405619244112, −9.813132539758024597169888458094, −9.066189793075234107295096885690, −7.967017025517771098830697611375, −7.26569317579054567255770987474, −5.93172699425919030021950594286, −5.19300327966827695169554689757, −4.01036689106736926473861118481, −2.47520088288264753241437176258, −2.23595805020013558835760984666,
2.23595805020013558835760984666, 2.47520088288264753241437176258, 4.01036689106736926473861118481, 5.19300327966827695169554689757, 5.93172699425919030021950594286, 7.26569317579054567255770987474, 7.967017025517771098830697611375, 9.066189793075234107295096885690, 9.813132539758024597169888458094, 10.57153349817273593405619244112