| L(s) = 1 | + 20.2·5-s − 24.7·7-s + 18.8·11-s − 48.4·13-s + 40.4·17-s + 7.82·19-s + 157.·23-s + 283.·25-s + 219.·29-s − 139.·31-s − 500.·35-s + 270.·37-s + 30.7·41-s + 57.2·43-s − 143.·47-s + 269.·49-s + 180.·53-s + 380.·55-s − 317.·59-s + 759.·61-s − 979.·65-s + 428.·67-s + 29.0·71-s − 327.·73-s − 465.·77-s − 1.02e3·79-s + 454.·83-s + ⋯ |
| L(s) = 1 | + 1.80·5-s − 1.33·7-s + 0.515·11-s − 1.03·13-s + 0.577·17-s + 0.0945·19-s + 1.43·23-s + 2.27·25-s + 1.40·29-s − 0.807·31-s − 2.41·35-s + 1.20·37-s + 0.117·41-s + 0.203·43-s − 0.444·47-s + 0.784·49-s + 0.466·53-s + 0.932·55-s − 0.699·59-s + 1.59·61-s − 1.86·65-s + 0.780·67-s + 0.0486·71-s − 0.525·73-s − 0.688·77-s − 1.45·79-s + 0.601·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.542476870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.542476870\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 20.2T + 125T^{2} \) |
| 7 | \( 1 + 24.7T + 343T^{2} \) |
| 11 | \( 1 - 18.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.82T + 6.85e3T^{2} \) |
| 23 | \( 1 - 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 139.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 30.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 57.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 143.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 180.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 317.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 759.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 428.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 29.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 327.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 454.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 677.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 397.T + 9.12e5T^{2} \) |
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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
−9.834433184048256046321795114980, −9.594156723697967982449615148613, −8.766512380412236135838003643053, −7.19528865790507042127703820160, −6.50428061032345454197879644447, −5.75789488173605486065169435552, −4.82628251683087474493083912980, −3.19901530076047260603952004370, −2.37444954348123779720200403292, −0.961708932992569776379382974742,
0.961708932992569776379382974742, 2.37444954348123779720200403292, 3.19901530076047260603952004370, 4.82628251683087474493083912980, 5.75789488173605486065169435552, 6.50428061032345454197879644447, 7.19528865790507042127703820160, 8.766512380412236135838003643053, 9.594156723697967982449615148613, 9.834433184048256046321795114980
