| L(s) = 1 | + 312.·5-s + 1.40e3·7-s − 6.20e3·11-s − 6.65e3·13-s + 1.09e4·17-s + 2.14e4·19-s − 3.08e4·23-s + 1.92e4·25-s − 8.57e4·29-s + 2.78e5·31-s + 4.38e5·35-s + 2.61e5·37-s + 4.37e5·41-s + 6.47e4·43-s + 1.25e6·47-s + 1.15e6·49-s + 1.05e6·53-s − 1.93e6·55-s − 1.93e6·59-s − 2.99e6·61-s − 2.07e6·65-s − 3.33e5·67-s + 4.41e6·71-s − 3.04e6·73-s − 8.71e6·77-s + 7.22e6·79-s + 2.36e6·83-s + ⋯ |
| L(s) = 1 | + 1.11·5-s + 1.54·7-s − 1.40·11-s − 0.840·13-s + 0.542·17-s + 0.717·19-s − 0.528·23-s + 0.246·25-s − 0.652·29-s + 1.67·31-s + 1.72·35-s + 0.847·37-s + 0.992·41-s + 0.124·43-s + 1.76·47-s + 1.39·49-s + 0.977·53-s − 1.56·55-s − 1.22·59-s − 1.68·61-s − 0.937·65-s − 0.135·67-s + 1.46·71-s − 0.916·73-s − 2.17·77-s + 1.64·79-s + 0.454·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.333362875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.333362875\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 312.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.40e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.20e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.65e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.09e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.14e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.08e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.57e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.78e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.61e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.37e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.47e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.25e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.05e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.93e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.99e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.33e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.41e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.04e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.22e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.42e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.63e6T + 8.07e13T^{2} \) |
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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
−10.07755402152953041721644816056, −9.185709462309167839402173419337, −7.909216145107617343434506653722, −7.57755339759149208677623844879, −5.95223906308124387362126338631, −5.28311885763958340054020860762, −4.50032833879486724624262380326, −2.70341546861458976024785246263, −1.99518262478147840621686975247, −0.843326171197254907551983689229,
0.843326171197254907551983689229, 1.99518262478147840621686975247, 2.70341546861458976024785246263, 4.50032833879486724624262380326, 5.28311885763958340054020860762, 5.95223906308124387362126338631, 7.57755339759149208677623844879, 7.909216145107617343434506653722, 9.185709462309167839402173419337, 10.07755402152953041721644816056
