| L(s) = 1 | + 2.65·2-s + 2.41·3-s + 5.05·4-s − 5-s + 6.40·6-s − 2.14·7-s + 8.10·8-s + 2.81·9-s − 2.65·10-s + 4.94·11-s + 12.1·12-s − 4.54·13-s − 5.70·14-s − 2.41·15-s + 11.4·16-s − 0.0358·17-s + 7.48·18-s − 5.05·20-s − 5.17·21-s + 13.1·22-s + 0.737·23-s + 19.5·24-s + 25-s − 12.0·26-s − 0.439·27-s − 10.8·28-s + 7.02·29-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 1.39·3-s + 2.52·4-s − 0.447·5-s + 2.61·6-s − 0.811·7-s + 2.86·8-s + 0.939·9-s − 0.839·10-s + 1.49·11-s + 3.51·12-s − 1.25·13-s − 1.52·14-s − 0.622·15-s + 2.85·16-s − 0.00869·17-s + 1.76·18-s − 1.12·20-s − 1.12·21-s + 2.80·22-s + 0.153·23-s + 3.99·24-s + 0.200·25-s − 2.36·26-s − 0.0845·27-s − 2.04·28-s + 1.30·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.767410490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.767410490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.65T + 2T^{2} \) |
| 3 | \( 1 - 2.41T + 3T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 + 0.0358T + 17T^{2} \) |
| 23 | \( 1 - 0.737T + 23T^{2} \) |
| 29 | \( 1 - 7.02T + 29T^{2} \) |
| 31 | \( 1 + 7.77T + 31T^{2} \) |
| 37 | \( 1 + 3.53T + 37T^{2} \) |
| 41 | \( 1 + 2.88T + 41T^{2} \) |
| 43 | \( 1 - 4.28T + 43T^{2} \) |
| 47 | \( 1 + 1.66T + 47T^{2} \) |
| 53 | \( 1 + 6.01T + 53T^{2} \) |
| 59 | \( 1 - 8.55T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 0.166T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 4.62T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 - 3.15T + 89T^{2} \) |
| 97 | \( 1 + 3.28T + 97T^{2} \) |
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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
−9.292274112055562238274963314913, −8.358476002974486019292930970503, −7.28451037371245290201088476983, −6.91876382944417661858690375035, −6.02889365857063202921945547592, −4.88023205234692816408586259284, −4.08049587743345078750588549359, −3.41677591342421968464749603418, −2.82780512370617744140575282410, −1.81474848050861129905345706513,
1.81474848050861129905345706513, 2.82780512370617744140575282410, 3.41677591342421968464749603418, 4.08049587743345078750588549359, 4.88023205234692816408586259284, 6.02889365857063202921945547592, 6.91876382944417661858690375035, 7.28451037371245290201088476983, 8.358476002974486019292930970503, 9.292274112055562238274963314913
