| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 4·7-s − 8-s + 9-s + 10-s + 6·11-s + 12-s − 4·13-s + 4·14-s − 15-s + 16-s + 6·17-s − 18-s − 20-s − 4·21-s − 6·22-s + 6·23-s − 24-s + 25-s + 4·26-s + 27-s − 4·28-s + 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s + 0.288·12-s − 1.10·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.223·20-s − 0.872·21-s − 1.27·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s − 0.755·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494429605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494429605\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−16.61837087870359, −16.12577650065411, −15.36066154760472, −14.88207672212260, −14.37594328184876, −13.80473700499483, −12.91290838304143, −12.31723132736317, −12.08806585016795, −11.35493158145793, −10.39961580490658, −9.972872108224414, −9.321079340493947, −9.091149335773776, −8.369683615595407, −7.352177422288140, −7.191588509536596, −6.486651519843896, −5.815705076669247, −4.761028105706978, −3.824541841713489, −3.275251286133736, −2.730631676664387, −1.512986528536166, −0.6443895030195491,
0.6443895030195491, 1.512986528536166, 2.730631676664387, 3.275251286133736, 3.824541841713489, 4.761028105706978, 5.815705076669247, 6.486651519843896, 7.191588509536596, 7.352177422288140, 8.369683615595407, 9.091149335773776, 9.321079340493947, 9.972872108224414, 10.39961580490658, 11.35493158145793, 12.08806585016795, 12.31723132736317, 12.91290838304143, 13.80473700499483, 14.37594328184876, 14.88207672212260, 15.36066154760472, 16.12577650065411, 16.61837087870359
