| L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s + 2·13-s − 15-s + 2·17-s + 4·23-s + 25-s − 27-s + 6·29-s − 4·31-s + 4·33-s − 6·37-s − 2·39-s − 10·41-s + 4·43-s + 45-s − 12·47-s − 7·49-s − 2·51-s + 6·53-s − 4·55-s − 12·59-s + 2·61-s + 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.696·33-s − 0.986·37-s − 0.320·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.75·47-s − 49-s − 0.280·51-s + 0.824·53-s − 0.539·55-s − 1.56·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.77711479706090, −12.35093510166900, −11.91785995287093, −11.36572639920434, −10.86773184493215, −10.59564035219110, −10.12455219307558, −9.744224396513745, −9.195520120460597, −8.583632509386848, −8.253965519580463, −7.739519148444873, −7.145818707990325, −6.713786064659241, −6.308272430239916, −5.612787835380646, −5.392637599407813, −4.820885586410729, −4.510660997994715, −3.573769599834181, −3.178550079761797, −2.714963351666588, −1.831833484838075, −1.509394035386220, −0.6939207875199140, 0,
0.6939207875199140, 1.509394035386220, 1.831833484838075, 2.714963351666588, 3.178550079761797, 3.573769599834181, 4.510660997994715, 4.820885586410729, 5.392637599407813, 5.612787835380646, 6.308272430239916, 6.713786064659241, 7.145818707990325, 7.739519148444873, 8.253965519580463, 8.583632509386848, 9.195520120460597, 9.744224396513745, 10.12455219307558, 10.59564035219110, 10.86773184493215, 11.36572639920434, 11.91785995287093, 12.35093510166900, 12.77711479706090
