| L(s) = 1 | − 27·3-s + 103.·5-s + 1.54e3·7-s + 729·9-s + 2.05e3·11-s + 2.19e3·13-s − 2.78e3·15-s + 1.57e4·17-s + 287.·19-s − 4.17e4·21-s − 4.01e4·23-s − 6.75e4·25-s − 1.96e4·27-s + 8.73e4·29-s − 1.00e5·31-s − 5.53e4·33-s + 1.59e5·35-s + 2.25e5·37-s − 5.93e4·39-s + 2.39e5·41-s − 4.88e5·43-s + 7.51e4·45-s + 1.15e6·47-s + 1.56e6·49-s − 4.24e5·51-s − 1.46e5·53-s + 2.11e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.368·5-s + 1.70·7-s + 0.333·9-s + 0.464·11-s + 0.277·13-s − 0.212·15-s + 0.776·17-s + 0.00963·19-s − 0.983·21-s − 0.687·23-s − 0.864·25-s − 0.192·27-s + 0.665·29-s − 0.604·31-s − 0.268·33-s + 0.627·35-s + 0.731·37-s − 0.160·39-s + 0.543·41-s − 0.936·43-s + 0.122·45-s + 1.62·47-s + 1.90·49-s − 0.448·51-s − 0.135·53-s + 0.171·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.455583895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.455583895\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 5 | \( 1 - 103.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.54e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.05e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 1.57e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 287.T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.01e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 8.73e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.00e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.25e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.39e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.88e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.15e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.46e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.43e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.02e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.24e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.07e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.53e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.54e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.85e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.33e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.27e6T + 8.07e13T^{2} \) |
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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
−11.57432994145866941956827456227, −10.78672016189597953343552516396, −9.702403660456319654739071496441, −8.398798032232476722320759917511, −7.49550770443543554076067224490, −6.06801674344421169504107702454, −5.14163555191958925715392864308, −4.01688737595682335170003403655, −2.01384654427283587139073159985, −0.983245570027881767846114462054,
0.983245570027881767846114462054, 2.01384654427283587139073159985, 4.01688737595682335170003403655, 5.14163555191958925715392864308, 6.06801674344421169504107702454, 7.49550770443543554076067224490, 8.398798032232476722320759917511, 9.702403660456319654739071496441, 10.78672016189597953343552516396, 11.57432994145866941956827456227
