| L(s) = 1 | + 10.1·5-s + 29.1·7-s + 19.5·11-s + 60.1·13-s + 102.·17-s + 113.·19-s − 14.8·23-s − 21.4·25-s − 68.1·29-s + 158.·31-s + 296.·35-s − 75.8·37-s − 364.·41-s − 291.·43-s + 454.·47-s + 508.·49-s − 560.·53-s + 198.·55-s + 359.·59-s + 383.·61-s + 612.·65-s − 736.·67-s − 360.·71-s − 1.01e3·73-s + 569.·77-s + 592.·79-s + 231.·83-s + ⋯ |
| L(s) = 1 | + 0.910·5-s + 1.57·7-s + 0.535·11-s + 1.28·13-s + 1.46·17-s + 1.37·19-s − 0.134·23-s − 0.171·25-s − 0.436·29-s + 0.920·31-s + 1.43·35-s − 0.336·37-s − 1.38·41-s − 1.03·43-s + 1.41·47-s + 1.48·49-s − 1.45·53-s + 0.487·55-s + 0.792·59-s + 0.803·61-s + 1.16·65-s − 1.34·67-s − 0.602·71-s − 1.61·73-s + 0.843·77-s + 0.843·79-s + 0.305·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.009517226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.009517226\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 10.1T + 125T^{2} \) |
| 7 | \( 1 - 29.1T + 343T^{2} \) |
| 11 | \( 1 - 19.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 113.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 14.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 68.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 158.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 75.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 364.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 291.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 454.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 560.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 359.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 383.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 736.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 360.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 592.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 231.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 472.T + 9.12e5T^{2} \) |
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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
−9.313044607207130713659480139693, −8.381304378534260556122253889295, −7.84006312459919360732546333023, −6.79687110780454979653843406580, −5.68220853430738012977334977898, −5.33123063187051831503448494545, −4.14766203234155087515126307363, −3.09527695360003947584973962079, −1.60090226685674035153945986285, −1.24259206601098229504536875904,
1.24259206601098229504536875904, 1.60090226685674035153945986285, 3.09527695360003947584973962079, 4.14766203234155087515126307363, 5.33123063187051831503448494545, 5.68220853430738012977334977898, 6.79687110780454979653843406580, 7.84006312459919360732546333023, 8.381304378534260556122253889295, 9.313044607207130713659480139693
