| L(s) = 1 | − 3·3-s + 9.82·5-s + 9·9-s − 14.1·11-s − 26.1·13-s − 29.4·15-s − 78.5·17-s + 73.1·19-s − 96·23-s − 28.4·25-s − 27·27-s + 173.·29-s + 67.2·31-s + 42.5·33-s − 301.·37-s + 78.3·39-s + 472.·41-s − 463.·43-s + 88.4·45-s − 91.1·47-s + 235.·51-s − 163.·53-s − 139.·55-s − 219.·57-s − 600.·59-s + 571.·61-s − 256.·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.878·5-s + 0.333·9-s − 0.388·11-s − 0.557·13-s − 0.507·15-s − 1.12·17-s + 0.883·19-s − 0.870·23-s − 0.227·25-s − 0.192·27-s + 1.10·29-s + 0.389·31-s + 0.224·33-s − 1.34·37-s + 0.321·39-s + 1.79·41-s − 1.64·43-s + 0.292·45-s − 0.283·47-s + 0.647·51-s − 0.423·53-s − 0.341·55-s − 0.510·57-s − 1.32·59-s + 1.19·61-s − 0.489·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 9.82T + 125T^{2} \) |
| 11 | \( 1 + 14.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 78.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 73.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96T + 1.21e4T^{2} \) |
| 29 | \( 1 - 173.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 67.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 472.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 463.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 91.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 163.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 600.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 571.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 539.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 442.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 45.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 686.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 660.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 658.T + 9.12e5T^{2} \) |
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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
−9.978754787555343134085683454326, −9.164599289499779074575478207739, −8.047569945955533689784127294872, −6.98673762847700125848783934190, −6.13918555541203747343483691575, −5.29034577231318367990490530520, −4.37490867951764857594552708454, −2.78990180525466626535874346812, −1.62445215325896474259516055578, 0,
1.62445215325896474259516055578, 2.78990180525466626535874346812, 4.37490867951764857594552708454, 5.29034577231318367990490530520, 6.13918555541203747343483691575, 6.98673762847700125848783934190, 8.047569945955533689784127294872, 9.164599289499779074575478207739, 9.978754787555343134085683454326
