| L(s) = 1 | − 8·3-s + 10·5-s + 37·9-s − 40·11-s + 50·13-s − 80·15-s + 30·17-s − 40·19-s + 48·23-s − 25·25-s − 80·27-s − 34·29-s − 320·31-s + 320·33-s + 310·37-s − 400·39-s − 410·41-s + 152·43-s + 370·45-s + 416·47-s − 240·51-s − 410·53-s − 400·55-s + 320·57-s + 200·59-s − 30·61-s + 500·65-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 0.894·5-s + 1.37·9-s − 1.09·11-s + 1.06·13-s − 1.37·15-s + 0.428·17-s − 0.482·19-s + 0.435·23-s − 1/5·25-s − 0.570·27-s − 0.217·29-s − 1.85·31-s + 1.68·33-s + 1.37·37-s − 1.64·39-s − 1.56·41-s + 0.539·43-s + 1.22·45-s + 1.29·47-s − 0.658·51-s − 1.06·53-s − 0.980·55-s + 0.743·57-s + 0.441·59-s − 0.0629·61-s + 0.954·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 5 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 34 T + p^{3} T^{2} \) |
| 31 | \( 1 + 320 T + p^{3} T^{2} \) |
| 37 | \( 1 - 310 T + p^{3} T^{2} \) |
| 41 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 152 T + p^{3} T^{2} \) |
| 47 | \( 1 - 416 T + p^{3} T^{2} \) |
| 53 | \( 1 + 410 T + p^{3} T^{2} \) |
| 59 | \( 1 - 200 T + p^{3} T^{2} \) |
| 61 | \( 1 + 30 T + p^{3} T^{2} \) |
| 67 | \( 1 - 776 T + p^{3} T^{2} \) |
| 71 | \( 1 - 400 T + p^{3} T^{2} \) |
| 73 | \( 1 - 630 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1120 T + p^{3} T^{2} \) |
| 83 | \( 1 + 552 T + p^{3} T^{2} \) |
| 89 | \( 1 - 326 T + p^{3} T^{2} \) |
| 97 | \( 1 - 110 T + p^{3} T^{2} \) |
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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
−8.750552705929246819645170852438, −7.72301016659493264917537353240, −6.80532335971148512500743131162, −5.93899147897248115895260163155, −5.59987855878924136973767172598, −4.82965839427915963522177492473, −3.64336367323553873329985237829, −2.23740432192224094394071666895, −1.14028814389029285615172211207, 0,
1.14028814389029285615172211207, 2.23740432192224094394071666895, 3.64336367323553873329985237829, 4.82965839427915963522177492473, 5.59987855878924136973767172598, 5.93899147897248115895260163155, 6.80532335971148512500743131162, 7.72301016659493264917537353240, 8.750552705929246819645170852438
