| L(s) = 1 | − 4.03·2-s + 3·3-s + 8.24·4-s − 8.08·5-s − 12.0·6-s − 5.95·7-s − 0.978·8-s + 9·9-s + 32.5·10-s − 17.2·11-s + 24.7·12-s + 23.9·14-s − 24.2·15-s − 61.9·16-s + 92.9·17-s − 36.2·18-s − 13.3·19-s − 66.6·20-s − 17.8·21-s + 69.4·22-s + 219.·23-s − 2.93·24-s − 59.5·25-s + 27·27-s − 49.0·28-s − 199.·29-s + 97.7·30-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 0.577·3-s + 1.03·4-s − 0.723·5-s − 0.822·6-s − 0.321·7-s − 0.0432·8-s + 0.333·9-s + 1.03·10-s − 0.472·11-s + 0.594·12-s + 0.457·14-s − 0.417·15-s − 0.968·16-s + 1.32·17-s − 0.474·18-s − 0.161·19-s − 0.745·20-s − 0.185·21-s + 0.673·22-s + 1.99·23-s − 0.0249·24-s − 0.476·25-s + 0.192·27-s − 0.331·28-s − 1.27·29-s + 0.595·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 4.03T + 8T^{2} \) |
| 5 | \( 1 + 8.08T + 125T^{2} \) |
| 7 | \( 1 + 5.95T + 343T^{2} \) |
| 11 | \( 1 + 17.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 92.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 13.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 219.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 199.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 307.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 333.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 200.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 116.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 26.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 280.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 207.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 285.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 317.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 63.0T + 3.89e5T^{2} \) |
| 79 | \( 1 + 623.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 659.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.27e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 603.T + 9.12e5T^{2} \) |
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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
−9.659118468522740944389873996107, −9.308794630191387338265787500615, −8.209503492055228736877888454286, −7.65601465803357551848010980380, −6.97592881170867078943900571295, −5.41094768324995984441411885311, −3.97017568211379918568550628465, −2.82321121770087945833686076265, −1.33821019167064680746293540103, 0,
1.33821019167064680746293540103, 2.82321121770087945833686076265, 3.97017568211379918568550628465, 5.41094768324995984441411885311, 6.97592881170867078943900571295, 7.65601465803357551848010980380, 8.209503492055228736877888454286, 9.308794630191387338265787500615, 9.659118468522740944389873996107
