| L(s) = 1 | − 2·5-s − 3·9-s − 4·11-s − 13-s + 17-s − 4·19-s − 4·23-s − 25-s − 6·29-s − 10·37-s − 2·41-s − 4·43-s + 6·45-s + 8·47-s − 7·49-s + 10·53-s + 8·55-s − 12·59-s + 2·61-s + 2·65-s − 12·67-s − 2·73-s − 4·79-s + 9·81-s − 12·83-s − 2·85-s − 14·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s − 1.20·11-s − 0.277·13-s + 0.242·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s − 1.64·37-s − 0.312·41-s − 0.609·43-s + 0.894·45-s + 1.16·47-s − 49-s + 1.37·53-s + 1.07·55-s − 1.56·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s − 0.234·73-s − 0.450·79-s + 81-s − 1.31·83-s − 0.216·85-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−16.68380841554688, −16.11965856589845, −15.44079378024214, −15.18735813520514, −14.55052018781467, −13.85146034367770, −13.43414553262555, −12.60795081656856, −12.21871060425911, −11.59369106275017, −11.10293817056440, −10.44714593477555, −10.02800696561383, −9.058872757448903, −8.525972151120064, −8.011095907492476, −7.506411275509635, −6.860379720230827, −5.865094374396131, −5.524405932713946, −4.701357739188510, −3.967447373166145, −3.278515475855164, −2.561035980701246, −1.739959892205017, 0, 0,
1.739959892205017, 2.561035980701246, 3.278515475855164, 3.967447373166145, 4.701357739188510, 5.524405932713946, 5.865094374396131, 6.860379720230827, 7.506411275509635, 8.011095907492476, 8.525972151120064, 9.058872757448903, 10.02800696561383, 10.44714593477555, 11.10293817056440, 11.59369106275017, 12.21871060425911, 12.60795081656856, 13.43414553262555, 13.85146034367770, 14.55052018781467, 15.18735813520514, 15.44079378024214, 16.11965856589845, 16.68380841554688
