| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 3·7-s − 8-s + 9-s + 10-s + 11-s + 12-s − 3·14-s − 15-s + 16-s − 6·17-s − 18-s − 5·19-s − 20-s + 3·21-s − 22-s − 23-s − 24-s + 25-s + 27-s + 3·28-s − 6·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.801·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 1.14·19-s − 0.223·20-s + 0.654·21-s − 0.213·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.566·28-s − 1.11·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 157170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 157170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.49625916734069, −13.10828061786403, −12.57247136653982, −11.82689338796140, −11.69624223308895, −11.02582030887903, −10.67519686902105, −10.34076963746417, −9.507358719090384, −9.057825063812313, −8.741367003300991, −8.213601733912126, −7.939241897719246, −7.227481551746807, −6.971940797920027, −6.278708273816185, −5.760064042087453, −4.950464245429520, −4.406586548734119, −4.089752974175658, −3.378213449102499, −2.627463649841821, −1.941416362148451, −1.790667917976236, −0.8045965776331603, 0,
0.8045965776331603, 1.790667917976236, 1.941416362148451, 2.627463649841821, 3.378213449102499, 4.089752974175658, 4.406586548734119, 4.950464245429520, 5.760064042087453, 6.278708273816185, 6.971940797920027, 7.227481551746807, 7.939241897719246, 8.213601733912126, 8.741367003300991, 9.057825063812313, 9.507358719090384, 10.34076963746417, 10.67519686902105, 11.02582030887903, 11.69624223308895, 11.82689338796140, 12.57247136653982, 13.10828061786403, 13.49625916734069
