| L(s) = 1 | + 2·2-s + 4·4-s + 13.2·5-s − 10.2·7-s + 8·8-s + 26.4·10-s + 14.6·11-s − 23.2·13-s − 20.5·14-s + 16·16-s + 119.·17-s + 112.·19-s + 52.9·20-s + 29.3·22-s + 23·23-s + 50.0·25-s − 46.5·26-s − 41.1·28-s − 104.·29-s − 27.1·31-s + 32·32-s + 238.·34-s − 136.·35-s − 24.7·37-s + 224.·38-s + 105.·40-s − 174.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.18·5-s − 0.555·7-s + 0.353·8-s + 0.836·10-s + 0.402·11-s − 0.496·13-s − 0.392·14-s + 0.250·16-s + 1.70·17-s + 1.35·19-s + 0.591·20-s + 0.284·22-s + 0.208·23-s + 0.400·25-s − 0.351·26-s − 0.277·28-s − 0.670·29-s − 0.157·31-s + 0.176·32-s + 1.20·34-s − 0.657·35-s − 0.109·37-s + 0.960·38-s + 0.418·40-s − 0.663·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.759315437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.759315437\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 13.2T + 125T^{2} \) |
| 7 | \( 1 + 10.2T + 343T^{2} \) |
| 11 | \( 1 - 14.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 112.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 104.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 27.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 24.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 174.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 506.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 256.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 200.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 446.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 419.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 70.3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 190.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 480.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 407.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 985.T + 9.12e5T^{2} \) |
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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
−10.77900889568792293204920980083, −9.747275279061429552576988787686, −9.416629323362669803984168011436, −7.79232524202977853181969974793, −6.86381256384171263698049440017, −5.77190981391525465562046380516, −5.28350104519318577067199458421, −3.71438678960576077646538483439, −2.66693349339710316460825945911, −1.27587717155017220594931219924,
1.27587717155017220594931219924, 2.66693349339710316460825945911, 3.71438678960576077646538483439, 5.28350104519318577067199458421, 5.77190981391525465562046380516, 6.86381256384171263698049440017, 7.79232524202977853181969974793, 9.416629323362669803984168011436, 9.747275279061429552576988787686, 10.77900889568792293204920980083
