| L(s) = 1 | + 2-s − 1.82·3-s + 4-s + 5-s − 1.82·6-s − 3.99·7-s + 8-s + 0.337·9-s + 10-s − 4.89·11-s − 1.82·12-s − 3.99·14-s − 1.82·15-s + 16-s + 6.62·17-s + 0.337·18-s + 2.09·19-s + 20-s + 7.29·21-s − 4.89·22-s − 0.991·23-s − 1.82·24-s + 25-s + 4.86·27-s − 3.99·28-s + 5.84·29-s − 1.82·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.05·3-s + 0.5·4-s + 0.447·5-s − 0.745·6-s − 1.50·7-s + 0.353·8-s + 0.112·9-s + 0.316·10-s − 1.47·11-s − 0.527·12-s − 1.06·14-s − 0.471·15-s + 0.250·16-s + 1.60·17-s + 0.0795·18-s + 0.480·19-s + 0.223·20-s + 1.59·21-s − 1.04·22-s − 0.206·23-s − 0.372·24-s + 0.200·25-s + 0.936·27-s − 0.754·28-s + 1.08·29-s − 0.333·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.449367993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.449367993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.82T + 3T^{2} \) |
| 7 | \( 1 + 3.99T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 17 | \( 1 - 6.62T + 17T^{2} \) |
| 19 | \( 1 - 2.09T + 19T^{2} \) |
| 23 | \( 1 + 0.991T + 23T^{2} \) |
| 29 | \( 1 - 5.84T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 3.29T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 - 1.13T + 43T^{2} \) |
| 47 | \( 1 - 1.61T + 47T^{2} \) |
| 53 | \( 1 + 0.549T + 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 + 1.37T + 61T^{2} \) |
| 67 | \( 1 + 3.62T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 8.94T + 73T^{2} \) |
| 79 | \( 1 - 2.55T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 3.44T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−9.881401728981280168222604291813, −8.457989024520648704141755514022, −7.50067509389485919633029303216, −6.59925042680349627956568704711, −5.92004571655200442321439680467, −5.47918390087963518040628346017, −4.61979286272015655369688762682, −3.20191829771695781622903568170, −2.70869307753814897341389186336, −0.77026383870769113965897067199,
0.77026383870769113965897067199, 2.70869307753814897341389186336, 3.20191829771695781622903568170, 4.61979286272015655369688762682, 5.47918390087963518040628346017, 5.92004571655200442321439680467, 6.59925042680349627956568704711, 7.50067509389485919633029303216, 8.457989024520648704141755514022, 9.881401728981280168222604291813
