| L(s) = 1 | + 3.23·5-s − 3.23·7-s − 2·11-s + 13-s + 4.47·17-s + 0.763·19-s + 6.47·23-s + 5.47·25-s − 4.47·29-s + 5.70·31-s − 10.4·35-s + 8.47·37-s + 3.23·41-s − 2.47·43-s − 10.9·47-s + 3.47·49-s − 0.472·53-s − 6.47·55-s + 0.472·59-s + 3.52·61-s + 3.23·65-s + 11.2·67-s − 4.47·71-s − 8.47·73-s + 6.47·77-s − 8.94·79-s − 16.4·83-s + ⋯ |
| L(s) = 1 | + 1.44·5-s − 1.22·7-s − 0.603·11-s + 0.277·13-s + 1.08·17-s + 0.175·19-s + 1.34·23-s + 1.09·25-s − 0.830·29-s + 1.02·31-s − 1.77·35-s + 1.39·37-s + 0.505·41-s − 0.376·43-s − 1.59·47-s + 0.496·49-s − 0.0648·53-s − 0.872·55-s + 0.0614·59-s + 0.451·61-s + 0.401·65-s + 1.37·67-s − 0.530·71-s − 0.991·73-s + 0.737·77-s − 1.00·79-s − 1.80·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.426041622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.426041622\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 - 3.23T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 0.472T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 + 8.47T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−7.85940393100499585129625228657, −7.03589763751261290736639175346, −6.41336852849480342039274056469, −5.78621558179945390090068298879, −5.35128240239363582744719724629, −4.39973387194027321844592162394, −3.14691582792048410338680359738, −2.90824942597166494670352965013, −1.80513133874401553108041526298, −0.791489719075464007013768928105,
0.791489719075464007013768928105, 1.80513133874401553108041526298, 2.90824942597166494670352965013, 3.14691582792048410338680359738, 4.39973387194027321844592162394, 5.35128240239363582744719724629, 5.78621558179945390090068298879, 6.41336852849480342039274056469, 7.03589763751261290736639175346, 7.85940393100499585129625228657
