| L(s) = 1 | − 140·5-s − 1.00e3·7-s − 4.00e3·11-s − 2.23e3·13-s − 1.14e3·17-s − 1.40e4·19-s − 8.80e4·23-s − 5.85e4·25-s + 4.13e4·29-s + 1.83e5·31-s + 1.40e5·35-s − 1.37e5·37-s − 1.23e5·41-s − 6.38e5·43-s + 7.76e5·47-s + 1.78e5·49-s + 9.81e5·53-s + 5.60e5·55-s − 5.68e5·59-s + 1.00e6·61-s + 3.13e5·65-s + 1.08e6·67-s − 2.33e6·71-s + 6.03e5·73-s + 4.00e6·77-s − 1.83e6·79-s − 2.28e6·83-s + ⋯ |
| L(s) = 1 | − 0.500·5-s − 1.10·7-s − 0.907·11-s − 0.282·13-s − 0.0564·17-s − 0.468·19-s − 1.50·23-s − 0.749·25-s + 0.314·29-s + 1.10·31-s + 0.552·35-s − 0.446·37-s − 0.280·41-s − 1.22·43-s + 1.09·47-s + 0.216·49-s + 0.905·53-s + 0.454·55-s − 0.360·59-s + 0.566·61-s + 0.141·65-s + 0.440·67-s − 0.775·71-s + 0.181·73-s + 1.00·77-s − 0.418·79-s − 0.438·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6701769795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6701769795\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 140T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.00e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.00e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.23e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.14e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.40e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.80e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.13e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.83e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.37e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.23e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.76e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.81e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.68e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.00e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.08e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.33e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.03e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.83e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.28e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.18e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.61e6T + 8.07e13T^{2} \) |
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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
−10.33806393996286646712185246774, −9.881718908433774841145280452021, −8.582356295567239105163787896749, −7.73669457425568672347162690674, −6.66807788499226651518018091731, −5.70569028645574570526030057671, −4.38542963447861093875951981093, −3.32116391159899953978316574963, −2.20208813964733535194611677771, −0.37937693496838371459663408657,
0.37937693496838371459663408657, 2.20208813964733535194611677771, 3.32116391159899953978316574963, 4.38542963447861093875951981093, 5.70569028645574570526030057671, 6.66807788499226651518018091731, 7.73669457425568672347162690674, 8.582356295567239105163787896749, 9.881718908433774841145280452021, 10.33806393996286646712185246774
