| L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s − 4·7-s − 8-s + 9-s + 4·10-s + 11-s − 12-s + 2·13-s + 4·14-s + 4·15-s + 16-s − 6·17-s − 18-s + 4·19-s − 4·20-s + 4·21-s − 22-s − 6·23-s + 24-s + 11·25-s − 2·26-s − 27-s − 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s + 1.03·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.894·20-s + 0.872·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.03745526569070, −13.59174027804150, −12.74325927543917, −12.60736692631964, −12.08146838107093, −11.54427488822047, −11.14051659248537, −10.76481651022881, −10.25648350471149, −9.449784644962486, −9.101207009799403, −8.815948390940075, −7.873203629754132, −7.443431037545293, −7.307394570928806, −6.354436831802143, −6.226102865091428, −5.587948611387422, −4.439079204924389, −4.246073646722665, −3.550700845965442, −3.132152265534575, −2.345814733405462, −1.337887541070215, −0.4828344989779804, 0,
0.4828344989779804, 1.337887541070215, 2.345814733405462, 3.132152265534575, 3.550700845965442, 4.246073646722665, 4.439079204924389, 5.587948611387422, 6.226102865091428, 6.354436831802143, 7.307394570928806, 7.443431037545293, 7.873203629754132, 8.815948390940075, 9.101207009799403, 9.449784644962486, 10.25648350471149, 10.76481651022881, 11.14051659248537, 11.54427488822047, 12.08146838107093, 12.60736692631964, 12.74325927543917, 13.59174027804150, 14.03745526569070
