| L(s) = 1 | + 0.319·2-s − 7.89·4-s − 10.3·5-s + 16.6·7-s − 5.07·8-s − 3.29·10-s + 35.9·11-s + 53.3·13-s + 5.32·14-s + 61.5·16-s + 38.7·17-s + 108.·19-s + 81.5·20-s + 11.4·22-s + 5.01·23-s − 18.4·25-s + 17.0·26-s − 131.·28-s + 52.6·29-s − 118.·31-s + 60.2·32-s + 12.3·34-s − 172.·35-s + 278.·37-s + 34.5·38-s + 52.4·40-s + 86.6·41-s + ⋯ |
| L(s) = 1 | + 0.112·2-s − 0.987·4-s − 0.923·5-s + 0.900·7-s − 0.224·8-s − 0.104·10-s + 0.986·11-s + 1.13·13-s + 0.101·14-s + 0.961·16-s + 0.552·17-s + 1.30·19-s + 0.911·20-s + 0.111·22-s + 0.0454·23-s − 0.147·25-s + 0.128·26-s − 0.889·28-s + 0.337·29-s − 0.684·31-s + 0.333·32-s + 0.0624·34-s − 0.831·35-s + 1.23·37-s + 0.147·38-s + 0.207·40-s + 0.330·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.064330339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064330339\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 0.319T + 8T^{2} \) |
| 5 | \( 1 + 10.3T + 125T^{2} \) |
| 7 | \( 1 - 16.6T + 343T^{2} \) |
| 11 | \( 1 - 35.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 53.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 38.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 5.01T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 278.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 86.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 214.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 177.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 347.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 432.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 150.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 565.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 551.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 747.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 468.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 565.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 711.T + 9.12e5T^{2} \) |
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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
−8.649639672577842780163613473021, −8.006786777410834812821979735481, −7.45670614165401469730590597017, −6.26272648789195400088324290331, −5.41186862823467490961871715738, −4.58530329957804524214686129293, −3.85367405731060228335828763774, −3.27960195963277174723039781689, −1.47288119052954184035335249402, −0.72877845454553980390279168013,
0.72877845454553980390279168013, 1.47288119052954184035335249402, 3.27960195963277174723039781689, 3.85367405731060228335828763774, 4.58530329957804524214686129293, 5.41186862823467490961871715738, 6.26272648789195400088324290331, 7.45670614165401469730590597017, 8.006786777410834812821979735481, 8.649639672577842780163613473021
