| L(s) = 1 | − 1.77·2-s − 3.27·3-s + 1.15·4-s − 5-s + 5.81·6-s + 2.00·7-s + 1.49·8-s + 7.70·9-s + 1.77·10-s − 5.30·11-s − 3.78·12-s + 1.99·13-s − 3.55·14-s + 3.27·15-s − 4.97·16-s − 2.15·17-s − 13.6·18-s − 5.13·19-s − 1.15·20-s − 6.54·21-s + 9.42·22-s + 2.16·23-s − 4.90·24-s + 25-s − 3.53·26-s − 15.3·27-s + 2.31·28-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 1.88·3-s + 0.577·4-s − 0.447·5-s + 2.37·6-s + 0.756·7-s + 0.530·8-s + 2.56·9-s + 0.561·10-s − 1.59·11-s − 1.09·12-s + 0.552·13-s − 0.949·14-s + 0.844·15-s − 1.24·16-s − 0.523·17-s − 3.22·18-s − 1.17·19-s − 0.258·20-s − 1.42·21-s + 2.00·22-s + 0.452·23-s − 1.00·24-s + 0.200·25-s − 0.693·26-s − 2.96·27-s + 0.436·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 1.77T + 2T^{2} \) |
| 3 | \( 1 + 3.27T + 3T^{2} \) |
| 7 | \( 1 - 2.00T + 7T^{2} \) |
| 11 | \( 1 + 5.30T + 11T^{2} \) |
| 13 | \( 1 - 1.99T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 23 | \( 1 - 2.16T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 - 6.02T + 37T^{2} \) |
| 41 | \( 1 - 8.60T + 41T^{2} \) |
| 43 | \( 1 - 3.93T + 43T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 - 0.642T + 53T^{2} \) |
| 59 | \( 1 + 7.87T + 59T^{2} \) |
| 61 | \( 1 - 4.28T + 61T^{2} \) |
| 67 | \( 1 - 1.67T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 0.667T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 1.46T + 83T^{2} \) |
| 89 | \( 1 - 4.48T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
| 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
−8.739064769110579609954208958465, −7.84083103634672220539419537263, −7.44847776123706274330467996216, −6.46142157941364621842583981906, −5.64699714359746390885449923338, −4.74364909566151596711772675878, −4.26819526745798521354105528499, −2.19866169532459012029838157465, −0.969409116106717988825244143645, 0,
0.969409116106717988825244143645, 2.19866169532459012029838157465, 4.26819526745798521354105528499, 4.74364909566151596711772675878, 5.64699714359746390885449923338, 6.46142157941364621842583981906, 7.44847776123706274330467996216, 7.84083103634672220539419537263, 8.739064769110579609954208958465
