| L(s) = 1 | + 1.94·2-s − 2.74·3-s + 1.76·4-s + 0.683·5-s − 5.32·6-s − 0.456·8-s + 4.52·9-s + 1.32·10-s + 0.192·11-s − 4.83·12-s − 6.05·13-s − 1.87·15-s − 4.41·16-s + 1.81·17-s + 8.77·18-s − 19-s + 1.20·20-s + 0.373·22-s + 2.34·23-s + 1.25·24-s − 4.53·25-s − 11.7·26-s − 4.17·27-s − 4.90·29-s − 3.63·30-s − 4.62·31-s − 7.65·32-s + ⋯ |
| L(s) = 1 | + 1.37·2-s − 1.58·3-s + 0.882·4-s + 0.305·5-s − 2.17·6-s − 0.161·8-s + 1.50·9-s + 0.419·10-s + 0.0580·11-s − 1.39·12-s − 1.67·13-s − 0.483·15-s − 1.10·16-s + 0.439·17-s + 2.06·18-s − 0.229·19-s + 0.269·20-s + 0.0795·22-s + 0.487·23-s + 0.255·24-s − 0.906·25-s − 2.30·26-s − 0.802·27-s − 0.910·29-s − 0.664·30-s − 0.830·31-s − 1.35·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.94T + 2T^{2} \) |
| 3 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 - 0.683T + 5T^{2} \) |
| 11 | \( 1 - 0.192T + 11T^{2} \) |
| 13 | \( 1 + 6.05T + 13T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 4.90T + 29T^{2} \) |
| 31 | \( 1 + 4.62T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 8.35T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 8.32T + 59T^{2} \) |
| 61 | \( 1 - 6.04T + 61T^{2} \) |
| 67 | \( 1 + 9.96T + 67T^{2} \) |
| 71 | \( 1 - 4.43T + 71T^{2} \) |
| 73 | \( 1 - 3.53T + 73T^{2} \) |
| 79 | \( 1 + 3.94T + 79T^{2} \) |
| 83 | \( 1 - 8.38T + 83T^{2} \) |
| 89 | \( 1 - 5.73T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
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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
−9.995822629831294548656976639975, −9.009300521023058620084010083451, −7.36008672584226718086569411439, −6.79041676704717992701747770305, −5.73119580115046634259039214237, −5.33320892970809562464690766868, −4.63520253953485838296569308245, −3.54783725578543357579983370133, −2.08118055205098450247582187973, 0,
2.08118055205098450247582187973, 3.54783725578543357579983370133, 4.63520253953485838296569308245, 5.33320892970809562464690766868, 5.73119580115046634259039214237, 6.79041676704717992701747770305, 7.36008672584226718086569411439, 9.009300521023058620084010083451, 9.995822629831294548656976639975
