L(s) = 1 | + 1.25·2-s + 3.01·3-s − 0.420·4-s + 3.78·6-s − 3.72·7-s − 3.04·8-s + 6.07·9-s − 3.35·11-s − 1.26·12-s + 4.84·13-s − 4.68·14-s − 2.98·16-s + 2.67·17-s + 7.63·18-s − 11.2·21-s − 4.21·22-s − 1.87·23-s − 9.16·24-s + 6.08·26-s + 9.25·27-s + 1.56·28-s + 5.25·29-s + 3.11·31-s + 2.33·32-s − 10.1·33-s + 3.36·34-s − 2.55·36-s + ⋯ |
L(s) = 1 | + 0.888·2-s + 1.73·3-s − 0.210·4-s + 1.54·6-s − 1.40·7-s − 1.07·8-s + 2.02·9-s − 1.01·11-s − 0.365·12-s + 1.34·13-s − 1.25·14-s − 0.745·16-s + 0.648·17-s + 1.79·18-s − 2.44·21-s − 0.899·22-s − 0.391·23-s − 1.87·24-s + 1.19·26-s + 1.78·27-s + 0.295·28-s + 0.975·29-s + 0.560·31-s + 0.412·32-s − 1.75·33-s + 0.576·34-s − 0.425·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.594007753\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.594007753\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 1.25T + 2T^{2} \) |
| 3 | \( 1 - 3.01T + 3T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 23 | \( 1 + 1.87T + 23T^{2} \) |
| 29 | \( 1 - 5.25T + 29T^{2} \) |
| 31 | \( 1 - 3.11T + 31T^{2} \) |
| 37 | \( 1 - 0.992T + 37T^{2} \) |
| 41 | \( 1 + 0.416T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 + 2.24T + 47T^{2} \) |
| 53 | \( 1 - 0.260T + 53T^{2} \) |
| 59 | \( 1 - 5.18T + 59T^{2} \) |
| 61 | \( 1 - 0.768T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 2.00T + 71T^{2} \) |
| 73 | \( 1 + 4.56T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 - 7.66T + 89T^{2} \) |
| 97 | \( 1 + 7.22T + 97T^{2} \) |
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
−8.040518766998500345373806617857, −6.99168020946986363221622233431, −6.31323530425075468188112773665, −5.69135624093268232587133676731, −4.73183362291016980636891374677, −3.89150548880845549859795989769, −3.45833221866717842493540990610, −2.93143676117273412925634568013, −2.29155713407610786109714760554, −0.818733642924098121464851027166,
0.818733642924098121464851027166, 2.29155713407610786109714760554, 2.93143676117273412925634568013, 3.45833221866717842493540990610, 3.89150548880845549859795989769, 4.73183362291016980636891374677, 5.69135624093268232587133676731, 6.31323530425075468188112773665, 6.99168020946986363221622233431, 8.040518766998500345373806617857