| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 6·11-s − 12-s + 6·13-s + 14-s − 15-s + 16-s − 17-s − 18-s + 6·19-s + 20-s + 21-s − 6·22-s + 6·23-s + 24-s + 25-s − 6·26-s − 27-s − 28-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 + 1.80·11-s − 0.288·12-s + 1.66·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.218·21-s − 1.27·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.757517303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.757517303\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.63562567437381, −13.14369320461835, −12.65465512406251, −11.98563709599159, −11.57737137783467, −11.22767361546962, −10.79667006222685, −10.20322112840294, −9.560175047424054, −9.280085123860994, −8.909361679879732, −8.317777397215251, −7.682654035745839, −6.891230626099738, −6.687549208637810, −6.254709539567245, −5.660394531703729, −5.203832136699348, −4.291982342468172, −3.786466710391645, −3.252449502205915, −2.541221805201484, −1.537293524268390, −1.147271639249301, −0.7285598703238091,
0.7285598703238091, 1.147271639249301, 1.537293524268390, 2.541221805201484, 3.252449502205915, 3.786466710391645, 4.291982342468172, 5.203832136699348, 5.660394531703729, 6.254709539567245, 6.687549208637810, 6.891230626099738, 7.682654035745839, 8.317777397215251, 8.909361679879732, 9.280085123860994, 9.560175047424054, 10.20322112840294, 10.79667006222685, 11.22767361546962, 11.57737137783467, 11.98563709599159, 12.65465512406251, 13.14369320461835, 13.63562567437381
