| L(s) = 1 | + 3.05·3-s − 3.42·5-s + 2.67·7-s + 6.34·9-s + 2.12·11-s − 1.43·13-s − 10.4·15-s − 17-s − 4.85·19-s + 8.16·21-s − 9.15·23-s + 6.74·25-s + 10.2·27-s − 4.91·29-s − 5.96·31-s + 6.47·33-s − 9.15·35-s − 3.49·37-s − 4.37·39-s + 4.45·41-s − 9.79·43-s − 21.7·45-s − 12.3·47-s + 0.140·49-s − 3.05·51-s − 4.08·53-s − 7.26·55-s + ⋯ |
| L(s) = 1 | + 1.76·3-s − 1.53·5-s + 1.01·7-s + 2.11·9-s + 0.639·11-s − 0.397·13-s − 2.70·15-s − 0.242·17-s − 1.11·19-s + 1.78·21-s − 1.90·23-s + 1.34·25-s + 1.96·27-s − 0.912·29-s − 1.07·31-s + 1.12·33-s − 1.54·35-s − 0.573·37-s − 0.700·39-s + 0.696·41-s − 1.49·43-s − 3.23·45-s − 1.80·47-s + 0.0201·49-s − 0.427·51-s − 0.561·53-s − 0.979·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 3.05T + 3T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 + 1.43T + 13T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 + 9.15T + 23T^{2} \) |
| 29 | \( 1 + 4.91T + 29T^{2} \) |
| 31 | \( 1 + 5.96T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 - 4.45T + 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 4.08T + 53T^{2} \) |
| 61 | \( 1 - 8.60T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 7.95T + 73T^{2} \) |
| 79 | \( 1 + 8.63T + 79T^{2} \) |
| 83 | \( 1 - 4.37T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.77253204486810192417182273006, −7.16382142455720356294827250152, −6.39434092774693352529581987328, −5.02607873722602437211249712086, −4.30601851315833275655743262898, −3.83102478758924744253953450562, −3.33568468412394009955645279069, −2.12319218628916696859647709987, −1.69834128185127170096220065441, 0,
1.69834128185127170096220065441, 2.12319218628916696859647709987, 3.33568468412394009955645279069, 3.83102478758924744253953450562, 4.30601851315833275655743262898, 5.02607873722602437211249712086, 6.39434092774693352529581987328, 7.16382142455720356294827250152, 7.77253204486810192417182273006
