| L(s) = 1 | + 1.32·2-s + 3-s − 0.253·4-s − 2.02·5-s + 1.32·6-s − 2.97·8-s + 9-s − 2.67·10-s − 0.0680·11-s − 0.253·12-s + 13-s − 2.02·15-s − 3.42·16-s + 6.04·17-s + 1.32·18-s + 5.34·19-s + 0.513·20-s − 0.0899·22-s + 5.55·23-s − 2.97·24-s − 0.896·25-s + 1.32·26-s + 27-s − 2.38·29-s − 2.67·30-s + 3.17·31-s + 1.42·32-s + ⋯ |
| L(s) = 1 | + 0.934·2-s + 0.577·3-s − 0.126·4-s − 0.905·5-s + 0.539·6-s − 1.05·8-s + 0.333·9-s − 0.846·10-s − 0.0205·11-s − 0.0731·12-s + 0.277·13-s − 0.523·15-s − 0.857·16-s + 1.46·17-s + 0.311·18-s + 1.22·19-s + 0.114·20-s − 0.0191·22-s + 1.15·23-s − 0.607·24-s − 0.179·25-s + 0.259·26-s + 0.192·27-s − 0.442·29-s − 0.488·30-s + 0.570·31-s + 0.251·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.644453763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.644453763\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 11 | \( 1 + 0.0680T + 11T^{2} \) |
| 17 | \( 1 - 6.04T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 - 2.36T + 41T^{2} \) |
| 43 | \( 1 + 0.103T + 43T^{2} \) |
| 47 | \( 1 - 4.06T + 47T^{2} \) |
| 53 | \( 1 + 7.95T + 53T^{2} \) |
| 59 | \( 1 - 6.88T + 59T^{2} \) |
| 61 | \( 1 - 8.04T + 61T^{2} \) |
| 67 | \( 1 + 4.66T + 67T^{2} \) |
| 71 | \( 1 - 0.522T + 71T^{2} \) |
| 73 | \( 1 - 6.87T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 4.97T + 89T^{2} \) |
| 97 | \( 1 - 5.09T + 97T^{2} \) |
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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
−9.283878483441803939894467776105, −8.243956422241022988267329311486, −7.74638953709799384558019062933, −6.87130988522084236147750272925, −5.74167631239459292000252170711, −5.06922482594388236883415632511, −4.10315912295299274464107999607, −3.46329413596686342534483263780, −2.79464739626052682017834700317, −0.968811423314922648686546831147,
0.968811423314922648686546831147, 2.79464739626052682017834700317, 3.46329413596686342534483263780, 4.10315912295299274464107999607, 5.06922482594388236883415632511, 5.74167631239459292000252170711, 6.87130988522084236147750272925, 7.74638953709799384558019062933, 8.243956422241022988267329311486, 9.283878483441803939894467776105
