| L(s) = 1 | − 3·3-s − 7.65·5-s + 31.5·7-s + 9·9-s + 7.18·11-s + 84.3·13-s + 22.9·15-s + 17·17-s + 37.0·19-s − 94.7·21-s − 150.·23-s − 66.3·25-s − 27·27-s − 11.5·29-s + 53.2·31-s − 21.5·33-s − 241.·35-s − 99.2·37-s − 252.·39-s + 118.·41-s + 456.·43-s − 68.9·45-s − 571.·47-s + 654.·49-s − 51·51-s + 462.·53-s − 55.0·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.684·5-s + 1.70·7-s + 0.333·9-s + 0.197·11-s + 1.79·13-s + 0.395·15-s + 0.242·17-s + 0.447·19-s − 0.984·21-s − 1.36·23-s − 0.531·25-s − 0.192·27-s − 0.0741·29-s + 0.308·31-s − 0.113·33-s − 1.16·35-s − 0.440·37-s − 1.03·39-s + 0.450·41-s + 1.61·43-s − 0.228·45-s − 1.77·47-s + 1.90·49-s − 0.140·51-s + 1.19·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.016009002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.016009002\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 17 | \( 1 - 17T \) |
| good | 5 | \( 1 + 7.65T + 125T^{2} \) |
| 7 | \( 1 - 31.5T + 343T^{2} \) |
| 11 | \( 1 - 7.18T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.3T + 2.19e3T^{2} \) |
| 19 | \( 1 - 37.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 150.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 11.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 53.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 99.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 118.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 456.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 571.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 462.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 48.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 59.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 740.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 930.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 697.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 22.2T + 5.71e5T^{2} \) |
| 89 | \( 1 + 369.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−10.04535457794624609702385998385, −8.746613153990902947404937836719, −8.106663805415364892774812344185, −7.47375744207677805593469943175, −6.21135545867440453778243063513, −5.44515747847250181167662140658, −4.37191177252840728474099048608, −3.70239920650006194317725243307, −1.85330413527580777709638650853, −0.875155359528677262477399089313,
0.875155359528677262477399089313, 1.85330413527580777709638650853, 3.70239920650006194317725243307, 4.37191177252840728474099048608, 5.44515747847250181167662140658, 6.21135545867440453778243063513, 7.47375744207677805593469943175, 8.106663805415364892774812344185, 8.746613153990902947404937836719, 10.04535457794624609702385998385
