L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 2.25·7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 0.258·13-s + 2.25·14-s + 15-s + 16-s + 17-s + 18-s − 4.14·19-s + 20-s + 2.25·21-s + 22-s − 5.76·23-s + 24-s + 25-s − 0.258·26-s + 27-s + 2.25·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.853·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.0718·13-s + 0.603·14-s + 0.258·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 0.951·19-s + 0.223·20-s + 0.492·21-s + 0.213·22-s − 1.20·23-s + 0.204·24-s + 0.200·25-s − 0.0507·26-s + 0.192·27-s + 0.426·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.105837202\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.105837202\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 2.25T + 7T^{2} \) |
| 13 | \( 1 + 0.258T + 13T^{2} \) |
| 19 | \( 1 + 4.14T + 19T^{2} \) |
| 23 | \( 1 + 5.76T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 - 4.71T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 3.54T + 43T^{2} \) |
| 47 | \( 1 - 1.43T + 47T^{2} \) |
| 53 | \( 1 + 9.04T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 + 9.30T + 67T^{2} \) |
| 71 | \( 1 - 7.83T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 5.08T + 89T^{2} \) |
| 97 | \( 1 - 9.68T + 97T^{2} \) |
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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
−8.234691744843290136631655764359, −7.43508844398871467407813342791, −6.56927476883647991357736251279, −6.02504450253296380464944661325, −5.12600075049994523083944272280, −4.40749614557740349131429485287, −3.84665533452265178718053498890, −2.69766670758317421724021023760, −2.12791798608328171588820118731, −1.13789753752006385738872059112,
1.13789753752006385738872059112, 2.12791798608328171588820118731, 2.69766670758317421724021023760, 3.84665533452265178718053498890, 4.40749614557740349131429485287, 5.12600075049994523083944272280, 6.02504450253296380464944661325, 6.56927476883647991357736251279, 7.43508844398871467407813342791, 8.234691744843290136631655764359