| L(s) = 1 | − 3-s + 2·5-s + 9-s + 13-s − 2·15-s + 6·17-s − 4·19-s − 8·23-s − 25-s − 27-s − 10·29-s − 6·37-s − 39-s − 10·41-s + 4·43-s + 2·45-s + 8·47-s − 7·49-s − 6·51-s + 10·53-s + 4·57-s + 12·59-s + 14·61-s + 2·65-s + 12·67-s + 8·69-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.277·13-s − 0.516·15-s + 1.45·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.986·37-s − 0.160·39-s − 1.56·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s − 49-s − 0.840·51-s + 1.37·53-s + 0.529·57-s + 1.56·59-s + 1.79·61-s + 0.248·65-s + 1.46·67-s + 0.963·69-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.88751613807893, −12.50851416159660, −11.82122342122240, −11.74017674663030, −11.05540363622918, −10.52020400586110, −10.13427344237055, −9.820010470933045, −9.448610709931309, −8.699076082549111, −8.325356817250144, −7.828243345161674, −7.210021653595910, −6.809332344544512, −6.200081957835601, −5.783577857000459, −5.415288381577162, −5.137872428982507, −4.188156639925646, −3.724781485577871, −3.487728786244065, −2.243906760659086, −2.171587402866433, −1.501051073653332, −0.7593096177862281, 0,
0.7593096177862281, 1.501051073653332, 2.171587402866433, 2.243906760659086, 3.487728786244065, 3.724781485577871, 4.188156639925646, 5.137872428982507, 5.415288381577162, 5.783577857000459, 6.200081957835601, 6.809332344544512, 7.210021653595910, 7.828243345161674, 8.325356817250144, 8.699076082549111, 9.448610709931309, 9.820010470933045, 10.13427344237055, 10.52020400586110, 11.05540363622918, 11.74017674663030, 11.82122342122240, 12.50851416159660, 12.88751613807893
