| L(s) = 1 | − 5-s + 2·13-s − 2·17-s + 4·19-s − 8·23-s + 25-s − 2·29-s − 4·31-s + 6·37-s − 6·41-s + 4·43-s + 6·53-s + 6·61-s − 2·65-s − 4·67-s − 8·71-s − 10·73-s + 12·79-s − 4·83-s + 2·85-s − 6·89-s − 4·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 0.824·53-s + 0.768·61-s − 0.248·65-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 1.35·79-s − 0.439·83-s + 0.216·85-s − 0.635·89-s − 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.57187586221119, −13.20734573943660, −12.73405142472263, −12.07471593792240, −11.67542524978875, −11.44818143171190, −10.66277996654385, −10.43119757640801, −9.702524784543023, −9.356032119966243, −8.728452236504746, −8.301558610492490, −7.736003624141844, −7.386002721570022, −6.750469523688736, −6.226617672915681, −5.621697924296826, −5.303197408643538, −4.317092193078688, −4.169834214303611, −3.472674137016270, −2.935948765352968, −2.164819505661564, −1.598146226809860, −0.7805400272549499, 0,
0.7805400272549499, 1.598146226809860, 2.164819505661564, 2.935948765352968, 3.472674137016270, 4.169834214303611, 4.317092193078688, 5.303197408643538, 5.621697924296826, 6.226617672915681, 6.750469523688736, 7.386002721570022, 7.736003624141844, 8.301558610492490, 8.728452236504746, 9.356032119966243, 9.702524784543023, 10.43119757640801, 10.66277996654385, 11.44818143171190, 11.67542524978875, 12.07471593792240, 12.73405142472263, 13.20734573943660, 13.57187586221119
