| L(s) = 1 | + 114.·5-s − 1.43e3·7-s − 5.92e3·11-s − 1.14e4·13-s − 2.02e4·17-s + 6.35e3·19-s − 7.58e4·23-s − 6.50e4·25-s − 7.47e4·29-s + 1.89e5·31-s − 1.64e5·35-s − 3.34e4·37-s + 1.41e5·41-s + 2.46e5·43-s − 3.35e5·47-s + 1.24e6·49-s + 1.65e6·53-s − 6.76e5·55-s − 2.04e6·59-s − 5.90e5·61-s − 1.30e6·65-s − 5.35e4·67-s + 4.95e6·71-s + 8.17e5·73-s + 8.52e6·77-s − 7.57e6·79-s + 1.01e6·83-s + ⋯ |
| L(s) = 1 | + 0.408·5-s − 1.58·7-s − 1.34·11-s − 1.44·13-s − 0.998·17-s + 0.212·19-s − 1.29·23-s − 0.833·25-s − 0.569·29-s + 1.14·31-s − 0.647·35-s − 0.108·37-s + 0.320·41-s + 0.472·43-s − 0.470·47-s + 1.51·49-s + 1.52·53-s − 0.548·55-s − 1.29·59-s − 0.333·61-s − 0.590·65-s − 0.0217·67-s + 1.64·71-s + 0.245·73-s + 2.12·77-s − 1.72·79-s + 0.195·83-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
| \(L(4)\) |
\(\approx\) |
\(0.3485440495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3485440495\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 114.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.43e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.92e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.14e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.02e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 6.35e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.47e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.89e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.34e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.41e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.46e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.35e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.65e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.04e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.90e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 5.35e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.95e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 8.17e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.57e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.01e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.06e7T + 8.07e13T^{2} \) |
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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
−9.882268256790260996259945393221, −9.417123096991721609125370998457, −8.086194118367255530648913301876, −7.16953368899714330232328109379, −6.23101094600210386048578129155, −5.37359462189576261154947106116, −4.17016077633502235945807563156, −2.84276112405619116972240309816, −2.20970507292212183829634232154, −0.24401785642753182930883691537,
0.24401785642753182930883691537, 2.20970507292212183829634232154, 2.84276112405619116972240309816, 4.17016077633502235945807563156, 5.37359462189576261154947106116, 6.23101094600210386048578129155, 7.16953368899714330232328109379, 8.086194118367255530648913301876, 9.417123096991721609125370998457, 9.882268256790260996259945393221
