| L(s) = 1 | + 5-s + 7-s − 3·9-s + 6·13-s + 2·17-s − 4·19-s + 4·23-s + 25-s − 6·29-s + 35-s − 2·37-s + 6·41-s − 4·43-s − 3·45-s − 4·47-s + 49-s − 2·53-s − 12·59-s + 2·61-s − 3·63-s + 6·65-s + 8·67-s + 8·71-s + 10·73-s − 8·79-s + 9·81-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 9-s + 1.66·13-s + 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 0.169·35-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.447·45-s − 0.583·47-s + 1/7·49-s − 0.274·53-s − 1.56·59-s + 0.256·61-s − 0.377·63-s + 0.744·65-s + 0.977·67-s + 0.949·71-s + 1.17·73-s − 0.900·79-s + 81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.43669120460368, −13.90170346268341, −13.50330147000814, −12.92091032045751, −12.57555293088422, −11.81337530838000, −11.23214456159444, −10.96648647462546, −10.61591373147673, −9.800526829115689, −9.205383657712547, −8.860182851597154, −8.222747534679834, −7.989346175092930, −7.132194493394354, −6.429097797426393, −6.142356048201481, −5.448606579502432, −5.137626753914088, −4.236822889593268, −3.657849898707348, −3.100565983322345, −2.386816736686672, −1.645119305108351, −1.042129112636612, 0,
1.042129112636612, 1.645119305108351, 2.386816736686672, 3.100565983322345, 3.657849898707348, 4.236822889593268, 5.137626753914088, 5.448606579502432, 6.142356048201481, 6.429097797426393, 7.132194493394354, 7.989346175092930, 8.222747534679834, 8.860182851597154, 9.205383657712547, 9.800526829115689, 10.61591373147673, 10.96648647462546, 11.23214456159444, 11.81337530838000, 12.57555293088422, 12.92091032045751, 13.50330147000814, 13.90170346268341, 14.43669120460368
