| L(s) = 1 | + 2-s − 4-s − 5-s − 2·7-s − 3·8-s − 3·9-s − 10-s + 4·11-s + 2·13-s − 2·14-s − 16-s + 6·17-s − 3·18-s − 6·19-s + 20-s + 4·22-s − 6·23-s + 25-s + 2·26-s + 2·28-s + 2·29-s − 4·31-s + 5·32-s + 6·34-s + 2·35-s + 3·36-s − 10·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.755·7-s − 1.06·8-s − 9-s − 0.316·10-s + 1.20·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s + 1.45·17-s − 0.707·18-s − 1.37·19-s + 0.223·20-s + 0.852·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s + 0.377·28-s + 0.371·29-s − 0.718·31-s + 0.883·32-s + 1.02·34-s + 0.338·35-s + 1/2·36-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.151746511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.151746511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 1609 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.160495898992488010563908357711, −6.76932507839522203562434435338, −6.39957467664749223874626534497, −5.70483242942339885933399484058, −5.07752621324574494540157291505, −4.06039099860597644324637958431, −3.57887381547570279461282045178, −3.16283556428725777221114510278, −1.85483874348362810689279369983, −0.46533937243820610520634011810,
0.46533937243820610520634011810, 1.85483874348362810689279369983, 3.16283556428725777221114510278, 3.57887381547570279461282045178, 4.06039099860597644324637958431, 5.07752621324574494540157291505, 5.70483242942339885933399484058, 6.39957467664749223874626534497, 6.76932507839522203562434435338, 8.160495898992488010563908357711
