| L(s) = 1 | + 2-s + 3-s + 4-s + 2.51·5-s + 6-s + 8-s + 9-s + 2.51·10-s + 0.959·11-s + 12-s + 13-s + 2.51·15-s + 16-s + 1.04·17-s + 18-s + 2.51·19-s + 2.51·20-s + 0.959·22-s − 0.513·23-s + 24-s + 1.31·25-s + 26-s + 27-s + 1.62·29-s + 2.51·30-s − 3.55·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.12·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.794·10-s + 0.289·11-s + 0.288·12-s + 0.277·13-s + 0.648·15-s + 0.250·16-s + 0.252·17-s + 0.235·18-s + 0.576·19-s + 0.561·20-s + 0.204·22-s − 0.106·23-s + 0.204·24-s + 0.263·25-s + 0.196·26-s + 0.192·27-s + 0.302·29-s + 0.458·30-s − 0.638·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.025089173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.025089173\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2.51T + 5T^{2} \) |
| 11 | \( 1 - 0.959T + 11T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 19 | \( 1 - 2.51T + 19T^{2} \) |
| 23 | \( 1 + 0.513T + 23T^{2} \) |
| 29 | \( 1 - 1.62T + 29T^{2} \) |
| 31 | \( 1 + 3.55T + 31T^{2} \) |
| 37 | \( 1 - 0.270T + 37T^{2} \) |
| 41 | \( 1 + 4.96T + 41T^{2} \) |
| 43 | \( 1 - 6.59T + 43T^{2} \) |
| 47 | \( 1 - 5.79T + 47T^{2} \) |
| 53 | \( 1 - 1.86T + 53T^{2} \) |
| 59 | \( 1 + 3.55T + 59T^{2} \) |
| 61 | \( 1 + 3.23T + 61T^{2} \) |
| 67 | \( 1 - 2.38T + 67T^{2} \) |
| 71 | \( 1 + 4.49T + 71T^{2} \) |
| 73 | \( 1 + 6.51T + 73T^{2} \) |
| 79 | \( 1 + 0.102T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 + 2.91T + 97T^{2} \) |
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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
−8.568679881237646476114166191509, −7.61241438393790778548380402218, −6.97524276328558954768734263596, −6.06597217875352164658349205820, −5.61478199870822337784478972126, −4.70050572464254485559564836935, −3.81294922013416390282203023435, −2.98802147868224673421632770228, −2.13608093885835813816521444730, −1.28180011609979738357738958396,
1.28180011609979738357738958396, 2.13608093885835813816521444730, 2.98802147868224673421632770228, 3.81294922013416390282203023435, 4.70050572464254485559564836935, 5.61478199870822337784478972126, 6.06597217875352164658349205820, 6.97524276328558954768734263596, 7.61241438393790778548380402218, 8.568679881237646476114166191509
