| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 2·11-s + 12-s + 14-s − 15-s + 16-s − 3·17-s + 18-s + 6·19-s − 20-s + 21-s − 2·22-s − 4·23-s + 24-s − 4·25-s + 27-s + 28-s − 5·29-s − 30-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 − 0.603·11-s + 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.188·28-s − 0.928·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15918 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15918 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 379 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.96788332200806, −15.59111207736416, −15.21027750483281, −14.55234368700922, −13.82669380544073, −13.72359031634310, −13.03905498048801, −12.29185385565622, −11.96212415623008, −11.12138045065008, −10.94699653116192, −9.817486070782547, −9.676160300340744, −8.658584795620960, −8.064295061277329, −7.634140172302024, −7.079628301579976, −6.258594268174998, −5.564847591252937, −4.898479167340667, −4.288611739877671, −3.587573246929597, −2.987721728536927, −2.165575879169169, −1.431746109063120, 0,
1.431746109063120, 2.165575879169169, 2.987721728536927, 3.587573246929597, 4.288611739877671, 4.898479167340667, 5.564847591252937, 6.258594268174998, 7.079628301579976, 7.634140172302024, 8.064295061277329, 8.658584795620960, 9.676160300340744, 9.817486070782547, 10.94699653116192, 11.12138045065008, 11.96212415623008, 12.29185385565622, 13.03905498048801, 13.72359031634310, 13.82669380544073, 14.55234368700922, 15.21027750483281, 15.59111207736416, 15.96788332200806
