| L(s) = 1 | + 2.02·2-s + 2.10·4-s − 3.48·5-s + 1.37·7-s + 0.213·8-s − 7.05·10-s − 0.191·11-s + 4.24·13-s + 2.77·14-s − 3.77·16-s − 4.07·17-s − 5.16·19-s − 7.32·20-s − 0.388·22-s − 0.978·23-s + 7.11·25-s + 8.59·26-s + 2.88·28-s − 10.0·29-s − 4.04·31-s − 8.08·32-s − 8.26·34-s − 4.76·35-s + 2.15·37-s − 10.4·38-s − 0.744·40-s − 0.0317·41-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.05·4-s − 1.55·5-s + 0.517·7-s + 0.0755·8-s − 2.23·10-s − 0.0577·11-s + 1.17·13-s + 0.741·14-s − 0.944·16-s − 0.989·17-s − 1.18·19-s − 1.63·20-s − 0.0827·22-s − 0.204·23-s + 1.42·25-s + 1.68·26-s + 0.545·28-s − 1.85·29-s − 0.727·31-s − 1.42·32-s − 1.41·34-s − 0.806·35-s + 0.353·37-s − 1.69·38-s − 0.117·40-s − 0.00496·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 5 | \( 1 + 3.48T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 + 0.191T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 4.07T + 17T^{2} \) |
| 19 | \( 1 + 5.16T + 19T^{2} \) |
| 23 | \( 1 + 0.978T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 4.04T + 31T^{2} \) |
| 37 | \( 1 - 2.15T + 37T^{2} \) |
| 41 | \( 1 + 0.0317T + 41T^{2} \) |
| 43 | \( 1 - 2.77T + 43T^{2} \) |
| 47 | \( 1 + 0.680T + 47T^{2} \) |
| 53 | \( 1 + 4.91T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 - 3.63T + 61T^{2} \) |
| 67 | \( 1 - 5.37T + 67T^{2} \) |
| 71 | \( 1 - 8.95T + 71T^{2} \) |
| 73 | \( 1 + 7.76T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 2.22T + 83T^{2} \) |
| 89 | \( 1 - 9.94T + 89T^{2} \) |
| 97 | \( 1 + 4.84T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.530870157944931102819483038222, −7.84217838752792714741724178034, −6.96365506708676985243235252067, −6.20412731309315540232461804618, −5.31823844500458410180651922202, −4.29411094829731234065507109263, −4.04762707759668056951119204519, −3.22183158209565315723152687716, −1.94638282209047600344721312281, 0,
1.94638282209047600344721312281, 3.22183158209565315723152687716, 4.04762707759668056951119204519, 4.29411094829731234065507109263, 5.31823844500458410180651922202, 6.20412731309315540232461804618, 6.96365506708676985243235252067, 7.84217838752792714741724178034, 8.530870157944931102819483038222
