| L(s) = 1 | − 8.30·2-s − 58.9·4-s + 187.·5-s − 1.50e3·7-s + 1.55e3·8-s − 1.55e3·10-s − 1.94e3·11-s − 4.43e3·13-s + 1.24e4·14-s − 5.36e3·16-s − 3.18e4·17-s + 8.14e3·19-s − 1.10e4·20-s + 1.61e4·22-s − 1.19e4·23-s − 4.30e4·25-s + 3.68e4·26-s + 8.84e4·28-s + 1.15e5·29-s − 1.00e5·31-s − 1.54e5·32-s + 2.64e5·34-s − 2.81e5·35-s − 4.55e5·37-s − 6.76e4·38-s + 2.90e5·40-s − 3.96e5·41-s + ⋯ |
| L(s) = 1 | − 0.734·2-s − 0.460·4-s + 0.670·5-s − 1.65·7-s + 1.07·8-s − 0.492·10-s − 0.439·11-s − 0.559·13-s + 1.21·14-s − 0.327·16-s − 1.57·17-s + 0.272·19-s − 0.308·20-s + 0.323·22-s − 0.204·23-s − 0.551·25-s + 0.411·26-s + 0.761·28-s + 0.878·29-s − 0.604·31-s − 0.832·32-s + 1.15·34-s − 1.10·35-s − 1.47·37-s − 0.200·38-s + 0.718·40-s − 0.898·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.2045992570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2045992570\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 - 7.95e4T \) |
| good | 2 | \( 1 + 8.30T + 128T^{2} \) |
| 5 | \( 1 - 187.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.50e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.94e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.43e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.18e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 8.14e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.19e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.15e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.00e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.55e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.96e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 2.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.63e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 7.05e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.31e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.27e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.30e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.27e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.41e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.46e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.16e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.54e7T + 8.07e13T^{2} \) |
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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
−10.07102282770727902586905860961, −9.255962649182549903517486011442, −8.656934492980521006181066344777, −7.33047712810979147558436484574, −6.52493202125075789333327679181, −5.43337116604237895307634545298, −4.25856621714434512981635422099, −2.97756194082070389063364746335, −1.80446062168701081489564074899, −0.22635270048182646430057730836,
0.22635270048182646430057730836, 1.80446062168701081489564074899, 2.97756194082070389063364746335, 4.25856621714434512981635422099, 5.43337116604237895307634545298, 6.52493202125075789333327679181, 7.33047712810979147558436484574, 8.656934492980521006181066344777, 9.255962649182549903517486011442, 10.07102282770727902586905860961
