| L(s) = 1 | + 2-s + 4-s + 0.267·5-s − 0.732·7-s + 8-s + 0.267·10-s + 4.73·11-s − 0.732·14-s + 16-s + 2.26·17-s − 1.26·19-s + 0.267·20-s + 4.73·22-s + 6.19·23-s − 4.92·25-s − 0.732·28-s − 2.46·29-s + 5.46·31-s + 32-s + 2.26·34-s − 0.196·35-s − 10.4·37-s − 1.26·38-s + 0.267·40-s + 11.3·41-s + 7.66·43-s + 4.73·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.119·5-s − 0.276·7-s + 0.353·8-s + 0.0847·10-s + 1.42·11-s − 0.195·14-s + 0.250·16-s + 0.550·17-s − 0.290·19-s + 0.0599·20-s + 1.00·22-s + 1.29·23-s − 0.985·25-s − 0.138·28-s − 0.457·29-s + 0.981·31-s + 0.176·32-s + 0.388·34-s − 0.0331·35-s − 1.72·37-s − 0.205·38-s + 0.0423·40-s + 1.77·41-s + 1.16·43-s + 0.713·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.309113222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.309113222\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.267T + 5T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 - 5.46T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 7.66T + 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 + 0.464T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 1.19T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 - 9.73T + 73T^{2} \) |
| 79 | \( 1 + 9.46T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.777285779634056065292063414770, −7.84798522162871798466678426427, −6.98057238369366006138231477940, −6.42472535630975994459228731397, −5.68561993460932933947777542085, −4.81707916508546336497260291380, −3.92345352723761991098957054046, −3.31062698000435634896133731644, −2.17333292392493369761009921253, −1.06160160747160921100454593550,
1.06160160747160921100454593550, 2.17333292392493369761009921253, 3.31062698000435634896133731644, 3.92345352723761991098957054046, 4.81707916508546336497260291380, 5.68561993460932933947777542085, 6.42472535630975994459228731397, 6.98057238369366006138231477940, 7.84798522162871798466678426427, 8.777285779634056065292063414770
