L(s) = 1 | + 1.56·2-s + 0.438·4-s − 5-s + 7-s − 2.43·8-s − 1.56·10-s + 5.56·13-s + 1.56·14-s − 4.68·16-s − 4.12·17-s + 6·19-s − 0.438·20-s + 4·23-s − 4·25-s + 8.68·26-s + 0.438·28-s − 6.68·29-s + 8.24·31-s − 2.43·32-s − 6.43·34-s − 35-s − 2.68·37-s + 9.36·38-s + 2.43·40-s + 7.56·41-s − 5.68·43-s + 6.24·46-s + ⋯ |
L(s) = 1 | + 1.10·2-s + 0.219·4-s − 0.447·5-s + 0.377·7-s − 0.862·8-s − 0.493·10-s + 1.54·13-s + 0.417·14-s − 1.17·16-s − 0.999·17-s + 1.37·19-s − 0.0980·20-s + 0.834·23-s − 0.800·25-s + 1.70·26-s + 0.0828·28-s − 1.24·29-s + 1.48·31-s − 0.431·32-s − 1.10·34-s − 0.169·35-s − 0.441·37-s + 1.51·38-s + 0.385·40-s + 1.18·41-s − 0.866·43-s + 0.920·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.117292828\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.117292828\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 13 | \( 1 - 5.56T + 13T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 + 2.68T + 37T^{2} \) |
| 41 | \( 1 - 7.56T + 41T^{2} \) |
| 43 | \( 1 + 5.68T + 43T^{2} \) |
| 47 | \( 1 + 3.43T + 47T^{2} \) |
| 53 | \( 1 + 7.80T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 13.3T + 61T^{2} \) |
| 67 | \( 1 - 8.80T + 67T^{2} \) |
| 71 | \( 1 + 3.12T + 71T^{2} \) |
| 73 | \( 1 - 7.12T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 - 8.80T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 1.31T + 97T^{2} \) |
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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
−7.898543690609411864746338023076, −6.99540824853919190190709927635, −6.30053615977820001508550809503, −5.68320767252609571009571367424, −4.95656109418086200624495290194, −4.33159726488713996562505696884, −3.59847011773255159057625182558, −3.11036956953458942711757073676, −1.94590455423141065286861026535, −0.75530878401955880664863734415,
0.75530878401955880664863734415, 1.94590455423141065286861026535, 3.11036956953458942711757073676, 3.59847011773255159057625182558, 4.33159726488713996562505696884, 4.95656109418086200624495290194, 5.68320767252609571009571367424, 6.30053615977820001508550809503, 6.99540824853919190190709927635, 7.898543690609411864746338023076