| L(s) = 1 | + 2-s + 3-s + 4-s + 3.10·5-s + 6-s − 4.91·7-s + 8-s + 9-s + 3.10·10-s + 4.33·11-s + 12-s − 1.81·13-s − 4.91·14-s + 3.10·15-s + 16-s − 0.710·17-s + 18-s − 0.578·19-s + 3.10·20-s − 4.91·21-s + 4.33·22-s − 7.62·23-s + 24-s + 4.62·25-s − 1.81·26-s + 27-s − 4.91·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.38·5-s + 0.408·6-s − 1.85·7-s + 0.353·8-s + 0.333·9-s + 0.981·10-s + 1.30·11-s + 0.288·12-s − 0.503·13-s − 1.31·14-s + 0.801·15-s + 0.250·16-s − 0.172·17-s + 0.235·18-s − 0.132·19-s + 0.693·20-s − 1.07·21-s + 0.924·22-s − 1.59·23-s + 0.204·24-s + 0.925·25-s − 0.355·26-s + 0.192·27-s − 0.929·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.530635010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.530635010\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 - 3.10T + 5T^{2} \) |
| 7 | \( 1 + 4.91T + 7T^{2} \) |
| 11 | \( 1 - 4.33T + 11T^{2} \) |
| 13 | \( 1 + 1.81T + 13T^{2} \) |
| 17 | \( 1 + 0.710T + 17T^{2} \) |
| 19 | \( 1 + 0.578T + 19T^{2} \) |
| 23 | \( 1 + 7.62T + 23T^{2} \) |
| 29 | \( 1 + 4.52T + 29T^{2} \) |
| 31 | \( 1 + 2.52T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 7.96T + 41T^{2} \) |
| 43 | \( 1 + 6.33T + 43T^{2} \) |
| 47 | \( 1 - 1.04T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 61 | \( 1 + 7.88T + 61T^{2} \) |
| 67 | \( 1 + 1.04T + 67T^{2} \) |
| 71 | \( 1 - 3.44T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 7.49T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−11.70644329201018643291161166558, −10.23781211602667117674777011334, −9.610745658557449975783395298783, −9.107418230100929653869906712825, −7.39691757990425392575661592816, −6.23106535666624528442706980695, −6.04001986795429614560544551302, −4.24081346334978724421600893837, −3.15922689431724128727544062184, −2.00883822083719968710730918728,
2.00883822083719968710730918728, 3.15922689431724128727544062184, 4.24081346334978724421600893837, 6.04001986795429614560544551302, 6.23106535666624528442706980695, 7.39691757990425392575661592816, 9.107418230100929653869906712825, 9.610745658557449975783395298783, 10.23781211602667117674777011334, 11.70644329201018643291161166558
