| L(s) = 1 | − 2-s + 2.77·3-s + 4-s + 1.35·5-s − 2.77·6-s + 7-s − 8-s + 4.67·9-s − 1.35·10-s + 0.777·11-s + 2.77·12-s + 4.89·13-s − 14-s + 3.75·15-s + 16-s + 2.66·17-s − 4.67·18-s + 1.35·20-s + 2.77·21-s − 0.777·22-s + 4.50·23-s − 2.77·24-s − 3.16·25-s − 4.89·26-s + 4.65·27-s + 28-s − 0.997·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.59·3-s + 0.5·4-s + 0.605·5-s − 1.13·6-s + 0.377·7-s − 0.353·8-s + 1.55·9-s − 0.427·10-s + 0.234·11-s + 0.799·12-s + 1.35·13-s − 0.267·14-s + 0.968·15-s + 0.250·16-s + 0.645·17-s − 1.10·18-s + 0.302·20-s + 0.604·21-s − 0.165·22-s + 0.939·23-s − 0.565·24-s − 0.633·25-s − 0.960·26-s + 0.895·27-s + 0.188·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.569846047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.569846047\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.77T + 3T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 11 | \( 1 - 0.777T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 - 2.66T + 17T^{2} \) |
| 23 | \( 1 - 4.50T + 23T^{2} \) |
| 29 | \( 1 + 0.997T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 + 5.59T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 3.93T + 47T^{2} \) |
| 53 | \( 1 - 7.53T + 53T^{2} \) |
| 59 | \( 1 + 3.36T + 59T^{2} \) |
| 61 | \( 1 + 3.75T + 61T^{2} \) |
| 67 | \( 1 - 1.09T + 67T^{2} \) |
| 71 | \( 1 + 8.04T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 2.44T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 2.32T + 97T^{2} \) |
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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
−8.406907654055964043643148693546, −7.77198127998442354734814870885, −7.06933430634502835998562346996, −6.24959482331101134253851934954, −5.44319101973115627140612673002, −4.26702832726055656051855732332, −3.43469943722147422929113288805, −2.78968161224200205789825989484, −1.80597693901045688690173594548, −1.20448760554143559379976142793,
1.20448760554143559379976142793, 1.80597693901045688690173594548, 2.78968161224200205789825989484, 3.43469943722147422929113288805, 4.26702832726055656051855732332, 5.44319101973115627140612673002, 6.24959482331101134253851934954, 7.06933430634502835998562346996, 7.77198127998442354734814870885, 8.406907654055964043643148693546
