| L(s) = 1 | − 3.16·3-s + 1.08·5-s + 1.30·7-s + 7.00·9-s + 11-s + 2.53·13-s − 3.42·15-s − 5.43·17-s − 19-s − 4.14·21-s − 3.87·23-s − 3.82·25-s − 12.6·27-s − 2.41·29-s − 3.03·31-s − 3.16·33-s + 1.41·35-s + 6.85·37-s − 8.03·39-s − 6.11·41-s + 2.95·43-s + 7.57·45-s + 12.0·47-s − 5.28·49-s + 17.1·51-s − 0.992·53-s + 1.08·55-s + ⋯ |
| L(s) = 1 | − 1.82·3-s + 0.484·5-s + 0.494·7-s + 2.33·9-s + 0.301·11-s + 0.704·13-s − 0.883·15-s − 1.31·17-s − 0.229·19-s − 0.903·21-s − 0.807·23-s − 0.765·25-s − 2.43·27-s − 0.448·29-s − 0.545·31-s − 0.550·33-s + 0.239·35-s + 1.12·37-s − 1.28·39-s − 0.954·41-s + 0.450·43-s + 1.12·45-s + 1.76·47-s − 0.755·49-s + 2.40·51-s − 0.136·53-s + 0.145·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3.16T + 3T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 + 5.43T + 17T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 + 2.41T + 29T^{2} \) |
| 31 | \( 1 + 3.03T + 31T^{2} \) |
| 37 | \( 1 - 6.85T + 37T^{2} \) |
| 41 | \( 1 + 6.11T + 41T^{2} \) |
| 43 | \( 1 - 2.95T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 0.992T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 5.82T + 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 - 2.44T + 71T^{2} \) |
| 73 | \( 1 - 5.84T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 - 7.01T + 83T^{2} \) |
| 89 | \( 1 + 9.13T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−8.190921772729331922003185731812, −7.22947025045920332175607156290, −6.54096001274804188173082153659, −5.91426814675664494463015886968, −5.44382198383745584182255231630, −4.44316294142005100967815380518, −3.96876363645154699780832653829, −2.17224957713772946388474040427, −1.30164856754113484809788743669, 0,
1.30164856754113484809788743669, 2.17224957713772946388474040427, 3.96876363645154699780832653829, 4.44316294142005100967815380518, 5.44382198383745584182255231630, 5.91426814675664494463015886968, 6.54096001274804188173082153659, 7.22947025045920332175607156290, 8.190921772729331922003185731812
