| L(s) = 1 | + 2-s − 3-s + 4-s + 2.23·5-s − 6-s − 7-s + 8-s − 2·9-s + 2.23·10-s − 0.236·11-s − 12-s + 3.23·13-s − 14-s − 2.23·15-s + 16-s − 5.23·17-s − 2·18-s + 3·19-s + 2.23·20-s + 21-s − 0.236·22-s − 1.76·23-s − 24-s + 3.23·26-s + 5·27-s − 28-s − 5.47·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.999·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s − 0.666·9-s + 0.707·10-s − 0.0711·11-s − 0.288·12-s + 0.897·13-s − 0.267·14-s − 0.577·15-s + 0.250·16-s − 1.26·17-s − 0.471·18-s + 0.688·19-s + 0.499·20-s + 0.218·21-s − 0.0503·22-s − 0.367·23-s − 0.204·24-s + 0.634·26-s + 0.962·27-s − 0.188·28-s − 1.01·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 + 5.47T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 - 3.70T + 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 - 2.47T + 89T^{2} \) |
| 97 | \( 1 + 8.70T + 97T^{2} \) |
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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
−7.44482984300991408381424594777, −6.74782560390204799926369824700, −6.08056118779065143879496890316, −5.62396144851045154871533048667, −5.11921809614326345200059999305, −4.02559724548440029065740793764, −3.31637797868113239237007190613, −2.33264220122667763813456872787, −1.55947115768518815396984644050, 0,
1.55947115768518815396984644050, 2.33264220122667763813456872787, 3.31637797868113239237007190613, 4.02559724548440029065740793764, 5.11921809614326345200059999305, 5.62396144851045154871533048667, 6.08056118779065143879496890316, 6.74782560390204799926369824700, 7.44482984300991408381424594777
