| L(s) = 1 | + 2.41·2-s − 1.41·3-s + 3.82·4-s − 5-s − 3.41·6-s − 4.82·7-s + 4.41·8-s − 0.999·9-s − 2.41·10-s − 3.41·11-s − 5.41·12-s − 11.6·14-s + 1.41·15-s + 2.99·16-s + 0.828·17-s − 2.41·18-s − 0.585·19-s − 3.82·20-s + 6.82·21-s − 8.24·22-s + 1.41·23-s − 6.24·24-s + 25-s + 5.65·27-s − 18.4·28-s − 5.65·29-s + 3.41·30-s + ⋯ |
| L(s) = 1 | + 1.70·2-s − 0.816·3-s + 1.91·4-s − 0.447·5-s − 1.39·6-s − 1.82·7-s + 1.56·8-s − 0.333·9-s − 0.763·10-s − 1.02·11-s − 1.56·12-s − 3.11·14-s + 0.365·15-s + 0.749·16-s + 0.200·17-s − 0.569·18-s − 0.134·19-s − 0.856·20-s + 1.49·21-s − 1.75·22-s + 0.294·23-s − 1.27·24-s + 0.200·25-s + 1.08·27-s − 3.49·28-s − 1.05·29-s + 0.623·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 + 0.585T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 + 1.75T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 - 2.48T + 53T^{2} \) |
| 59 | \( 1 + 1.75T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 7.65T + 97T^{2} \) |
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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
−10.15767672223446692403396422962, −9.026942848136500443879317012396, −7.58237203867389184223307986446, −6.73744656346224362208410858969, −5.97031552585204471714607698511, −5.46396933825867778629999014387, −4.38993345435487524521668471709, −3.33790417930147427701901169465, −2.69989485395194313129565717715, 0,
2.69989485395194313129565717715, 3.33790417930147427701901169465, 4.38993345435487524521668471709, 5.46396933825867778629999014387, 5.97031552585204471714607698511, 6.73744656346224362208410858969, 7.58237203867389184223307986446, 9.026942848136500443879317012396, 10.15767672223446692403396422962
