| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s + 4·11-s − 12-s + 14-s − 15-s + 16-s − 6·17-s + 18-s + 20-s − 21-s + 4·22-s − 8·23-s − 24-s + 25-s − 27-s + 28-s + 10·29-s − 30-s + 8·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.223·20-s − 0.218·21-s + 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s + 1.85·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.976758894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.976758894\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.97106461248450, −14.12923556578135, −13.86313321503972, −13.65137175562096, −12.67357702728886, −12.28611917014441, −11.87038022520575, −11.33374217451062, −10.86116006472825, −10.18631408648691, −9.800695547882574, −9.017939124897245, −8.463341914589898, −7.887221123374769, −6.989351902362328, −6.519291481632441, −6.226839039625917, −5.584776117066894, −4.805361252156922, −4.287207536213471, −3.989283881775662, −2.872984645250642, −2.234888987931887, −1.525350443122611, −0.7017545903986008,
0.7017545903986008, 1.525350443122611, 2.234888987931887, 2.872984645250642, 3.989283881775662, 4.287207536213471, 4.805361252156922, 5.584776117066894, 6.226839039625917, 6.519291481632441, 6.989351902362328, 7.887221123374769, 8.463341914589898, 9.017939124897245, 9.800695547882574, 10.18631408648691, 10.86116006472825, 11.33374217451062, 11.87038022520575, 12.28611917014441, 12.67357702728886, 13.65137175562096, 13.86313321503972, 14.12923556578135, 14.97106461248450
