| L(s) = 1 | − 430·5-s − 1.22e3·7-s + 3.16e3·11-s + 6.11e3·13-s + 1.62e4·17-s − 5.47e3·19-s − 1.57e3·23-s + 1.06e5·25-s − 1.22e5·29-s + 2.51e5·31-s + 5.26e5·35-s − 5.23e4·37-s + 3.19e5·41-s + 7.10e5·43-s − 2.84e5·47-s + 6.74e5·49-s − 2.96e5·53-s − 1.36e6·55-s + 8.97e5·59-s − 8.84e5·61-s − 2.63e6·65-s + 4.65e6·67-s + 2.71e6·71-s − 5.67e6·73-s − 3.87e6·77-s − 5.12e6·79-s + 1.56e6·83-s + ⋯ |
| L(s) = 1 | − 1.53·5-s − 1.34·7-s + 0.716·11-s + 0.772·13-s + 0.803·17-s − 0.183·19-s − 0.0270·23-s + 1.36·25-s − 0.935·29-s + 1.51·31-s + 2.07·35-s − 0.169·37-s + 0.723·41-s + 1.36·43-s − 0.399·47-s + 0.819·49-s − 0.273·53-s − 1.10·55-s + 0.568·59-s − 0.499·61-s − 1.18·65-s + 1.89·67-s + 0.898·71-s − 1.70·73-s − 0.966·77-s − 1.16·79-s + 0.300·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.084792003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.084792003\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 86 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1224 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3164 T + p^{7} T^{2} \) |
| 13 | \( 1 - 6118 T + p^{7} T^{2} \) |
| 17 | \( 1 - 16270 T + p^{7} T^{2} \) |
| 19 | \( 1 + 5476 T + p^{7} T^{2} \) |
| 23 | \( 1 + 1576 T + p^{7} T^{2} \) |
| 29 | \( 1 + 122838 T + p^{7} T^{2} \) |
| 31 | \( 1 - 251360 T + p^{7} T^{2} \) |
| 37 | \( 1 + 52338 T + p^{7} T^{2} \) |
| 41 | \( 1 - 319398 T + p^{7} T^{2} \) |
| 43 | \( 1 - 710788 T + p^{7} T^{2} \) |
| 47 | \( 1 + 284112 T + p^{7} T^{2} \) |
| 53 | \( 1 + 296062 T + p^{7} T^{2} \) |
| 59 | \( 1 - 897548 T + p^{7} T^{2} \) |
| 61 | \( 1 + 884810 T + p^{7} T^{2} \) |
| 67 | \( 1 - 4659692 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2710792 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5670854 T + p^{7} T^{2} \) |
| 79 | \( 1 + 5124176 T + p^{7} T^{2} \) |
| 83 | \( 1 - 1563556 T + p^{7} T^{2} \) |
| 89 | \( 1 + 11605674 T + p^{7} T^{2} \) |
| 97 | \( 1 - 10931618 T + p^{7} T^{2} \) |
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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
−12.97183726390896094318812442835, −12.09871582437048449153286581700, −11.13994741844772149060024607298, −9.754070617768425944542938184578, −8.520305799434237397311727316335, −7.31479006282189591292870340544, −6.13885829857675449203011625378, −4.11698596982195771006762999370, −3.23857687122335288736747191421, −0.69291130145493708924236017457,
0.69291130145493708924236017457, 3.23857687122335288736747191421, 4.11698596982195771006762999370, 6.13885829857675449203011625378, 7.31479006282189591292870340544, 8.520305799434237397311727316335, 9.754070617768425944542938184578, 11.13994741844772149060024607298, 12.09871582437048449153286581700, 12.97183726390896094318812442835
