| L(s) = 1 | − 5-s + 4·7-s − 4·11-s + 13-s − 6·17-s + 4·23-s + 25-s + 6·29-s − 8·31-s − 4·35-s − 2·37-s − 10·41-s − 4·43-s − 8·47-s + 9·49-s + 2·53-s + 4·55-s − 4·59-s + 14·61-s − 65-s − 12·67-s + 8·71-s − 10·73-s − 16·77-s + 4·83-s + 6·85-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 1.20·11-s + 0.277·13-s − 1.45·17-s + 0.834·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.676·35-s − 0.328·37-s − 1.56·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.274·53-s + 0.539·55-s − 0.520·59-s + 1.79·61-s − 0.124·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s − 1.82·77-s + 0.439·83-s + 0.650·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.180604414497567589384264482821, −7.23465965688715827868346322721, −6.70429518063638549972823831992, −5.48630724706287589415453698556, −4.94348202561904907438355497303, −4.39520349459478338579537402434, −3.32240915229117906302579134756, −2.33364735907848921793197115976, −1.47206787733481655449551975299, 0,
1.47206787733481655449551975299, 2.33364735907848921793197115976, 3.32240915229117906302579134756, 4.39520349459478338579537402434, 4.94348202561904907438355497303, 5.48630724706287589415453698556, 6.70429518063638549972823831992, 7.23465965688715827868346322721, 8.180604414497567589384264482821
