| L(s) = 1 | − 5-s − 3·7-s − 5·11-s − 5·17-s − 19-s + 23-s + 25-s + 9·29-s − 4·31-s + 3·35-s − 3·37-s + 41-s − 3·43-s + 8·47-s + 2·49-s − 10·53-s + 5·55-s − 3·59-s + 7·61-s − 9·67-s − 7·71-s + 10·73-s + 15·77-s − 16·79-s − 12·83-s + 5·85-s − 15·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s − 1.50·11-s − 1.21·17-s − 0.229·19-s + 0.208·23-s + 1/5·25-s + 1.67·29-s − 0.718·31-s + 0.507·35-s − 0.493·37-s + 0.156·41-s − 0.457·43-s + 1.16·47-s + 2/7·49-s − 1.37·53-s + 0.674·55-s − 0.390·59-s + 0.896·61-s − 1.09·67-s − 0.830·71-s + 1.17·73-s + 1.70·77-s − 1.80·79-s − 1.31·83-s + 0.542·85-s − 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.98016483221341, −14.14193356468691, −13.78241848931467, −13.14023748375960, −12.77021653639910, −12.52294323955306, −11.76804995389109, −11.18147278613258, −10.70594833620145, −10.21966394324413, −9.832004316487263, −9.049653734191630, −8.652059512236523, −8.130249384770013, −7.459464265414493, −6.981634699499793, −6.469889939620209, −5.896755430220122, −5.224816670213413, −4.631150243027681, −4.103121638654220, −3.286744731211173, −2.793668861061841, −2.326048047574123, −1.259976587528407, 0, 0,
1.259976587528407, 2.326048047574123, 2.793668861061841, 3.286744731211173, 4.103121638654220, 4.631150243027681, 5.224816670213413, 5.896755430220122, 6.469889939620209, 6.981634699499793, 7.459464265414493, 8.130249384770013, 8.652059512236523, 9.049653734191630, 9.832004316487263, 10.21966394324413, 10.70594833620145, 11.18147278613258, 11.76804995389109, 12.52294323955306, 12.77021653639910, 13.14023748375960, 13.78241848931467, 14.14193356468691, 14.98016483221341
