| 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 + 2·13-s − 14-s − 15-s + 16-s + 4·17-s + 18-s − 20-s − 21-s + 4·22-s − 6·23-s + 24-s + 25-s + 2·26-s + 27-s − 28-s − 8·29-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 + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.852·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s − 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.29893523635675, −13.70334403381063, −13.51813865631336, −12.85208231466336, −12.17242693772456, −12.05199517553073, −11.45810193778324, −10.90539642571980, −10.27369705039328, −9.748721833326591, −9.276650187415801, −8.707115758085825, −8.105709611205970, −7.623768836475686, −7.128210465765970, −6.489752923805866, −5.971177037952643, −5.577902416016596, −4.596974385784032, −4.186235877502140, −3.574877794526344, −3.329452348582316, −2.532538326005687, −1.665369194018968, −1.216216085905322, 0,
1.216216085905322, 1.665369194018968, 2.532538326005687, 3.329452348582316, 3.574877794526344, 4.186235877502140, 4.596974385784032, 5.577902416016596, 5.971177037952643, 6.489752923805866, 7.128210465765970, 7.623768836475686, 8.105709611205970, 8.707115758085825, 9.276650187415801, 9.748721833326591, 10.27369705039328, 10.90539642571980, 11.45810193778324, 12.05199517553073, 12.17242693772456, 12.85208231466336, 13.51813865631336, 13.70334403381063, 14.29893523635675
