| L(s) = 1 | + 0.801·2-s − 3·3-s − 7.35·4-s + 6.45·5-s − 2.40·6-s − 17.3·7-s − 12.3·8-s + 9·9-s + 5.17·10-s − 11·11-s + 22.0·12-s − 17.2·13-s − 13.9·14-s − 19.3·15-s + 48.9·16-s + 130.·17-s + 7.21·18-s + 56.4·19-s − 47.5·20-s + 52.1·21-s − 8.81·22-s − 27.3·23-s + 36.9·24-s − 83.2·25-s − 13.8·26-s − 27·27-s + 127.·28-s + ⋯ |
| L(s) = 1 | + 0.283·2-s − 0.577·3-s − 0.919·4-s + 0.577·5-s − 0.163·6-s − 0.939·7-s − 0.543·8-s + 0.333·9-s + 0.163·10-s − 0.301·11-s + 0.530·12-s − 0.367·13-s − 0.266·14-s − 0.333·15-s + 0.765·16-s + 1.86·17-s + 0.0944·18-s + 0.681·19-s − 0.531·20-s + 0.542·21-s − 0.0854·22-s − 0.247·23-s + 0.314·24-s − 0.666·25-s − 0.104·26-s − 0.192·27-s + 0.863·28-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
| \(L(2)\) |
\(\approx\) |
\(0.8976546652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8976546652\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.801T + 8T^{2} \) |
| 5 | \( 1 - 6.45T + 125T^{2} \) |
| 7 | \( 1 + 17.3T + 343T^{2} \) |
| 13 | \( 1 + 17.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 130.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 56.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 27.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 254.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 275.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 193.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 176.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 3.09T + 7.95e4T^{2} \) |
| 47 | \( 1 + 81.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 441.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 442.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 779.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 897.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 281.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 414.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−9.118861049490938125867641533672, −7.85707192544566666533678580395, −7.28272995035452428862482528725, −6.01327514794028877242632787547, −5.63892823933672037556827393439, −5.00097758312454866850973161722, −3.74717868731278506177832668289, −3.22222698791146482370685516656, −1.71237086288024678616377140714, −0.43209149961191240361753672272,
0.43209149961191240361753672272, 1.71237086288024678616377140714, 3.22222698791146482370685516656, 3.74717868731278506177832668289, 5.00097758312454866850973161722, 5.63892823933672037556827393439, 6.01327514794028877242632787547, 7.28272995035452428862482528725, 7.85707192544566666533678580395, 9.118861049490938125867641533672
