| L(s) = 1 | − 13.0·5-s + 31.0·7-s − 59.5·11-s + 62.8·13-s + 97.4·17-s − 19·19-s + 160.·23-s + 44.7·25-s − 300.·29-s − 62.6·31-s − 403.·35-s + 91.7·37-s − 394.·41-s − 395.·43-s + 515.·47-s + 618.·49-s + 402.·53-s + 775.·55-s + 229.·59-s + 16.8·61-s − 818.·65-s + 32.1·67-s + 141.·71-s + 667.·73-s − 1.84e3·77-s + 510.·79-s − 201.·83-s + ⋯ |
| L(s) = 1 | − 1.16·5-s + 1.67·7-s − 1.63·11-s + 1.34·13-s + 1.39·17-s − 0.229·19-s + 1.45·23-s + 0.357·25-s − 1.92·29-s − 0.363·31-s − 1.95·35-s + 0.407·37-s − 1.50·41-s − 1.40·43-s + 1.59·47-s + 1.80·49-s + 1.04·53-s + 1.90·55-s + 0.506·59-s + 0.0354·61-s − 1.56·65-s + 0.0585·67-s + 0.237·71-s + 1.07·73-s − 2.72·77-s + 0.726·79-s − 0.265·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.971261680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.971261680\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 5 | \( 1 + 13.0T + 125T^{2} \) |
| 7 | \( 1 - 31.0T + 343T^{2} \) |
| 11 | \( 1 + 59.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 97.4T + 4.91e3T^{2} \) |
| 23 | \( 1 - 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 300.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 62.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 91.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 394.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 395.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 515.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 402.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 16.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 32.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 141.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 667.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 510.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 201.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 346.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 490.T + 9.12e5T^{2} \) |
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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
−8.913916728548588843493767872073, −8.178029012461631698944337738754, −7.81195515511471209218001324722, −7.12184973748936206402079982632, −5.51075477259800427274134009718, −5.17187611653275635814356132679, −4.03822505416348605806280971364, −3.24469770699109232009609908308, −1.85293253204800527556008565162, −0.72006897211103942193611156447,
0.72006897211103942193611156447, 1.85293253204800527556008565162, 3.24469770699109232009609908308, 4.03822505416348605806280971364, 5.17187611653275635814356132679, 5.51075477259800427274134009718, 7.12184973748936206402079982632, 7.81195515511471209218001324722, 8.178029012461631698944337738754, 8.913916728548588843493767872073
