| L(s) = 1 | + 2.82·5-s + 4·7-s + 1.41·11-s + 2.82·13-s + 4·17-s + 7.07·19-s − 4·23-s + 3.00·25-s − 8.48·29-s + 8·31-s + 11.3·35-s + 2.82·37-s − 2·41-s − 4.24·43-s + 9·49-s − 2.82·53-s + 4.00·55-s + 4.24·59-s − 8.48·61-s + 8.00·65-s − 4.24·67-s + 4·71-s − 4·73-s + 5.65·77-s − 8·79-s − 9.89·83-s + 11.3·85-s + ⋯ |
| L(s) = 1 | + 1.26·5-s + 1.51·7-s + 0.426·11-s + 0.784·13-s + 0.970·17-s + 1.62·19-s − 0.834·23-s + 0.600·25-s − 1.57·29-s + 1.43·31-s + 1.91·35-s + 0.464·37-s − 0.312·41-s − 0.646·43-s + 1.28·49-s − 0.388·53-s + 0.539·55-s + 0.552·59-s − 1.08·61-s + 0.992·65-s − 0.518·67-s + 0.474·71-s − 0.468·73-s + 0.644·77-s − 0.900·79-s − 1.08·83-s + 1.22·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.582732830\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.582732830\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 4T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.191634051256763922920265097778, −7.77434265068322976554730881963, −6.85702960783565882330285140443, −5.81185404773107409185316806673, −5.59622905751577463944222981570, −4.73785378234598858101922724001, −3.80692100644937147128704607075, −2.78088359809617169396418931579, −1.61950616328305713839618913971, −1.29370670186172753093625243889,
1.29370670186172753093625243889, 1.61950616328305713839618913971, 2.78088359809617169396418931579, 3.80692100644937147128704607075, 4.73785378234598858101922724001, 5.59622905751577463944222981570, 5.81185404773107409185316806673, 6.85702960783565882330285140443, 7.77434265068322976554730881963, 8.191634051256763922920265097778
