| L(s) = 1 | − 2.66·3-s − 5·5-s + 32.5·7-s − 19.8·9-s + 51.2·11-s + 40.1·13-s + 13.3·15-s + 93.3·17-s + 61.9·19-s − 86.8·21-s − 23·23-s + 25·25-s + 125.·27-s − 160.·29-s + 295.·31-s − 136.·33-s − 162.·35-s + 401.·37-s − 107.·39-s + 310.·41-s − 506.·43-s + 99.4·45-s − 188.·47-s + 717.·49-s − 249.·51-s + 162.·53-s − 256.·55-s + ⋯ |
| L(s) = 1 | − 0.513·3-s − 0.447·5-s + 1.75·7-s − 0.736·9-s + 1.40·11-s + 0.857·13-s + 0.229·15-s + 1.33·17-s + 0.747·19-s − 0.902·21-s − 0.208·23-s + 0.200·25-s + 0.891·27-s − 1.02·29-s + 1.71·31-s − 0.721·33-s − 0.786·35-s + 1.78·37-s − 0.440·39-s + 1.18·41-s − 1.79·43-s + 0.329·45-s − 0.584·47-s + 2.09·49-s − 0.683·51-s + 0.421·53-s − 0.628·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.671127500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.671127500\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 2.66T + 27T^{2} \) |
| 7 | \( 1 - 32.5T + 343T^{2} \) |
| 11 | \( 1 - 51.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 61.9T + 6.85e3T^{2} \) |
| 29 | \( 1 + 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 295.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 401.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 310.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 506.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 188.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 162.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 206.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 856.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 435.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 132.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 675.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 802.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 657.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.58e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.54e3T + 9.12e5T^{2} \) |
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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
−8.725413747272650634266181213387, −8.075457780020193002486371745964, −7.52670781168933481160511378914, −6.31829892733550661496088096368, −5.70222710607669211413271339209, −4.80693039231582658939144036784, −4.05466469126570284301435703710, −3.02731871393078015279589709931, −1.49175751646686152870988467699, −0.908148351383024656368158127258,
0.908148351383024656368158127258, 1.49175751646686152870988467699, 3.02731871393078015279589709931, 4.05466469126570284301435703710, 4.80693039231582658939144036784, 5.70222710607669211413271339209, 6.31829892733550661496088096368, 7.52670781168933481160511378914, 8.075457780020193002486371745964, 8.725413747272650634266181213387
