| L(s) = 1 | + 2-s − 3-s − 4-s + 2·5-s − 6-s − 7-s − 3·8-s + 9-s + 2·10-s + 12-s − 14-s − 2·15-s − 16-s + 2·17-s + 18-s + 2·19-s − 2·20-s + 21-s + 6·23-s + 3·24-s − 25-s − 27-s + 28-s + 29-s − 2·30-s − 8·31-s + 5·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 0.267·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.458·19-s − 0.447·20-s + 0.218·21-s + 1.25·23-s + 0.612·24-s − 1/5·25-s − 0.192·27-s + 0.188·28-s + 0.185·29-s − 0.365·30-s − 1.43·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73689 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73689 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.151387034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.151387034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.05297833516586, −13.46851344518862, −13.14594464117187, −12.61580945768366, −12.38162262186430, −11.58288394647276, −11.25985177297295, −10.47271779914877, −10.09411928796701, −9.537556787985234, −9.060956410778189, −8.776007572763622, −7.785179719775445, −7.294334850138732, −6.680333644199521, −5.956428415447889, −5.770933054983044, −5.234823367486142, −4.686917039413818, −4.104969243489946, −3.326121355162309, −2.974096867790683, −2.055773425528415, −1.276869405170867, −0.4728652801150800,
0.4728652801150800, 1.276869405170867, 2.055773425528415, 2.974096867790683, 3.326121355162309, 4.104969243489946, 4.686917039413818, 5.234823367486142, 5.770933054983044, 5.956428415447889, 6.680333644199521, 7.294334850138732, 7.785179719775445, 8.776007572763622, 9.060956410778189, 9.537556787985234, 10.09411928796701, 10.47271779914877, 11.25985177297295, 11.58288394647276, 12.38162262186430, 12.61580945768366, 13.14594464117187, 13.46851344518862, 14.05297833516586
