| L(s) = 1 | + 0.0567·2-s + 3·3-s − 7.99·4-s − 17.4·5-s + 0.170·6-s + 20.8·7-s − 0.907·8-s + 9·9-s − 0.989·10-s − 11·11-s − 23.9·12-s − 10.0·13-s + 1.18·14-s − 52.3·15-s + 63.9·16-s − 73.8·17-s + 0.510·18-s + 93.4·19-s + 139.·20-s + 62.4·21-s − 0.624·22-s − 163.·23-s − 2.72·24-s + 179.·25-s − 0.572·26-s + 27·27-s − 166.·28-s + ⋯ |
| L(s) = 1 | + 0.0200·2-s + 0.577·3-s − 0.999·4-s − 1.56·5-s + 0.0115·6-s + 1.12·7-s − 0.0401·8-s + 0.333·9-s − 0.0313·10-s − 0.301·11-s − 0.577·12-s − 0.215·13-s + 0.0225·14-s − 0.900·15-s + 0.998·16-s − 1.05·17-s + 0.00668·18-s + 1.12·19-s + 1.55·20-s + 0.648·21-s − 0.00604·22-s − 1.48·23-s − 0.0231·24-s + 1.43·25-s − 0.00431·26-s + 0.192·27-s − 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 0.0567T + 8T^{2} \) |
| 5 | \( 1 + 17.4T + 125T^{2} \) |
| 7 | \( 1 - 20.8T + 343T^{2} \) |
| 13 | \( 1 + 10.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 73.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 93.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 54.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 181.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 174.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 503.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 165.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 174.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 485.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 198.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 54.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 194.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 260.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 268.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 931.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.04e3T + 9.12e5T^{2} \) |
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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
−8.180874053208856887147152749712, −7.894163639987488451254898512880, −7.30902727691989533065810523365, −5.87510639579074598987087037178, −4.73709665606675840934347213813, −4.35316611046243105456850641000, −3.63945585251669215139955964275, −2.49792442418372334861910593345, −1.07083889585200306978407844278, 0,
1.07083889585200306978407844278, 2.49792442418372334861910593345, 3.63945585251669215139955964275, 4.35316611046243105456850641000, 4.73709665606675840934347213813, 5.87510639579074598987087037178, 7.30902727691989533065810523365, 7.894163639987488451254898512880, 8.180874053208856887147152749712
