| L(s) = 1 | − 13.7·2-s + 41.9·3-s + 60.3·4-s − 575.·6-s − 165.·7-s + 928.·8-s − 429.·9-s + 31.8·11-s + 2.53e3·12-s − 5.67e3·13-s + 2.26e3·14-s − 2.04e4·16-s − 1.07e4·17-s + 5.90e3·18-s − 4.26e4·19-s − 6.92e3·21-s − 436.·22-s − 1.21e4·23-s + 3.89e4·24-s + 7.78e4·26-s − 1.09e5·27-s − 9.96e3·28-s + 1.02e5·29-s − 2.02e5·31-s + 1.62e5·32-s + 1.33e3·33-s + 1.47e5·34-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 0.896·3-s + 0.471·4-s − 1.08·6-s − 0.181·7-s + 0.640·8-s − 0.196·9-s + 0.00721·11-s + 0.422·12-s − 0.715·13-s + 0.220·14-s − 1.24·16-s − 0.531·17-s + 0.238·18-s − 1.42·19-s − 0.163·21-s − 0.00874·22-s − 0.208·23-s + 0.574·24-s + 0.868·26-s − 1.07·27-s − 0.0858·28-s + 0.779·29-s − 1.22·31-s + 0.874·32-s + 0.00646·33-s + 0.644·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6379391219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6379391219\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 + 13.7T + 128T^{2} \) |
| 3 | \( 1 - 41.9T + 2.18e3T^{2} \) |
| 7 | \( 1 + 165.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 31.8T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.67e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.07e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.26e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.02e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.02e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.65e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.99e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.21e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.16e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.57e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.24e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.82e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.03e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.12e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.82e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.42e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.92e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.76e7T + 8.07e13T^{2} \) |
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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
−9.320223234536303937866274589183, −8.802077996229035666469623523521, −8.108185850719272383230550324918, −7.32158243989904908538137226097, −6.32795857942878863559147725631, −4.88792486241207822413082370382, −3.82743297385484141782257716894, −2.53440892977004920443816943362, −1.82315721408711431274552541705, −0.37714351998491782479539464514,
0.37714351998491782479539464514, 1.82315721408711431274552541705, 2.53440892977004920443816943362, 3.82743297385484141782257716894, 4.88792486241207822413082370382, 6.32795857942878863559147725631, 7.32158243989904908538137226097, 8.108185850719272383230550324918, 8.802077996229035666469623523521, 9.320223234536303937866274589183
