| L(s) = 1 | − 42.5·3-s + 51.7·5-s − 1.21e3·7-s − 372.·9-s + 2.25e3·11-s + 2.28e3·13-s − 2.20e3·15-s − 3.80e4·17-s + 6.85e3·19-s + 5.16e4·21-s + 6.87e4·23-s − 7.54e4·25-s + 1.09e5·27-s + 2.33e5·29-s + 3.12e5·31-s − 9.60e4·33-s − 6.27e4·35-s + 1.27e5·37-s − 9.74e4·39-s − 8.76e4·41-s + 9.39e5·43-s − 1.92e4·45-s − 9.75e5·47-s + 6.46e5·49-s + 1.61e6·51-s − 4.69e5·53-s + 1.16e5·55-s + ⋯ |
| L(s) = 1 | − 0.910·3-s + 0.185·5-s − 1.33·7-s − 0.170·9-s + 0.511·11-s + 0.288·13-s − 0.168·15-s − 1.87·17-s + 0.229·19-s + 1.21·21-s + 1.17·23-s − 0.965·25-s + 1.06·27-s + 1.77·29-s + 1.88·31-s − 0.465·33-s − 0.247·35-s + 0.415·37-s − 0.263·39-s − 0.198·41-s + 1.80·43-s − 0.0315·45-s − 1.36·47-s + 0.784·49-s + 1.70·51-s − 0.432·53-s + 0.0946·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9431763421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9431763421\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 3 | \( 1 + 42.5T + 2.18e3T^{2} \) |
| 5 | \( 1 - 51.7T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.21e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.25e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.28e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.80e4T + 4.10e8T^{2} \) |
| 23 | \( 1 - 6.87e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.33e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.12e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.27e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.76e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.39e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.75e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.69e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.30e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.27e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.74e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.10e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.24e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.65e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.59e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.06e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.07e6T + 8.07e13T^{2} \) |
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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
−13.00469694051606894486275594907, −11.90591770038364537316538501758, −10.95945463313312407159884773126, −9.784972755885055792202034094002, −8.677400519915408231517192549566, −6.67574488800335884529867457285, −6.17320406072806216890235118344, −4.56415112347354762337298952788, −2.83971026732950195504515494061, −0.65529836743731433622917632871,
0.65529836743731433622917632871, 2.83971026732950195504515494061, 4.56415112347354762337298952788, 6.17320406072806216890235118344, 6.67574488800335884529867457285, 8.677400519915408231517192549566, 9.784972755885055792202034094002, 10.95945463313312407159884773126, 11.90591770038364537316538501758, 13.00469694051606894486275594907
