| L(s) = 1 | − 2.58·2-s − 1.47·3-s + 4.67·4-s + 5-s + 3.80·6-s + 4.00·7-s − 6.92·8-s − 0.831·9-s − 2.58·10-s − 5.75·11-s − 6.89·12-s + 1.77·13-s − 10.3·14-s − 1.47·15-s + 8.53·16-s + 2.14·18-s + 1.30·19-s + 4.67·20-s − 5.90·21-s + 14.8·22-s + 3.35·23-s + 10.1·24-s + 25-s − 4.58·26-s + 5.64·27-s + 18.7·28-s + 3.55·29-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.850·3-s + 2.33·4-s + 0.447·5-s + 1.55·6-s + 1.51·7-s − 2.44·8-s − 0.277·9-s − 0.817·10-s − 1.73·11-s − 1.98·12-s + 0.492·13-s − 2.76·14-s − 0.380·15-s + 2.13·16-s + 0.506·18-s + 0.298·19-s + 1.04·20-s − 1.28·21-s + 3.17·22-s + 0.700·23-s + 2.08·24-s + 0.200·25-s − 0.899·26-s + 1.08·27-s + 3.54·28-s + 0.659·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5598999251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5598999251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 + 1.47T + 3T^{2} \) |
| 7 | \( 1 - 4.00T + 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 - 1.77T + 13T^{2} \) |
| 19 | \( 1 - 1.30T + 19T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 - 3.55T + 29T^{2} \) |
| 31 | \( 1 + 3.15T + 31T^{2} \) |
| 37 | \( 1 + 1.85T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 9.60T + 43T^{2} \) |
| 47 | \( 1 + 9.45T + 47T^{2} \) |
| 53 | \( 1 + 0.883T + 53T^{2} \) |
| 59 | \( 1 - 1.33T + 59T^{2} \) |
| 61 | \( 1 + 6.38T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 6.79T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 5.70T + 97T^{2} \) |
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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
−9.520442412670141065238301203193, −8.586477707834715274370083607302, −8.104062343454227451778504800603, −7.43149642160823803086557947367, −6.45216484624075759992296142900, −5.48992082188070680569391444128, −4.94100776676584120742478827210, −2.85098171495233642497439292063, −1.87469009938638372717647988089, −0.73729622942790623573803322614,
0.73729622942790623573803322614, 1.87469009938638372717647988089, 2.85098171495233642497439292063, 4.94100776676584120742478827210, 5.48992082188070680569391444128, 6.45216484624075759992296142900, 7.43149642160823803086557947367, 8.104062343454227451778504800603, 8.586477707834715274370083607302, 9.520442412670141065238301203193
