L(s) = 1 | − 2.12·2-s − 3-s + 2.51·4-s + 5-s + 2.12·6-s − 2.76·7-s − 1.08·8-s + 9-s − 2.12·10-s − 5.65·11-s − 2.51·12-s − 1.69·13-s + 5.88·14-s − 15-s − 2.71·16-s − 3.25·17-s − 2.12·18-s − 4.88·19-s + 2.51·20-s + 2.76·21-s + 12.0·22-s + 5.29·23-s + 1.08·24-s + 25-s + 3.60·26-s − 27-s − 6.95·28-s + ⋯ |
L(s) = 1 | − 1.50·2-s − 0.577·3-s + 1.25·4-s + 0.447·5-s + 0.867·6-s − 1.04·7-s − 0.383·8-s + 0.333·9-s − 0.671·10-s − 1.70·11-s − 0.724·12-s − 0.470·13-s + 1.57·14-s − 0.258·15-s − 0.679·16-s − 0.790·17-s − 0.500·18-s − 1.12·19-s + 0.561·20-s + 0.604·21-s + 2.55·22-s + 1.10·23-s + 0.221·24-s + 0.200·25-s + 0.707·26-s − 0.192·27-s − 1.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 + 3.25T + 17T^{2} \) |
| 19 | \( 1 + 4.88T + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 1.71T + 29T^{2} \) |
| 31 | \( 1 - 6.93T + 31T^{2} \) |
| 37 | \( 1 + 2.99T + 37T^{2} \) |
| 41 | \( 1 - 3.44T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 - 0.298T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 2.80T + 59T^{2} \) |
| 61 | \( 1 - 7.75T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 5.00T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 9.33T + 97T^{2} \) |
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
−7.84942488224343356785777003177, −6.93846359925691582615216704777, −6.70813592225182314140169275237, −5.73046036034386290062455518589, −5.02801298203167074731219581690, −4.10189762167297569467613004829, −2.59545428308430332770643897485, −2.35663904175082314245774093065, −0.842106380240291329502369027908, 0,
0.842106380240291329502369027908, 2.35663904175082314245774093065, 2.59545428308430332770643897485, 4.10189762167297569467613004829, 5.02801298203167074731219581690, 5.73046036034386290062455518589, 6.70813592225182314140169275237, 6.93846359925691582615216704777, 7.84942488224343356785777003177