| L(s) = 1 | − 0.447·2-s − 7.79·4-s − 1.93·5-s − 8.14·7-s + 7.06·8-s + 0.863·10-s − 8.40·11-s + 3.64·14-s + 59.2·16-s + 52.1·17-s + 48.8·19-s + 15.0·20-s + 3.76·22-s − 88.9·23-s − 121.·25-s + 63.5·28-s − 191.·29-s + 115.·31-s − 83.0·32-s − 23.3·34-s + 15.7·35-s − 136.·37-s − 21.8·38-s − 13.6·40-s − 436.·41-s + 202.·43-s + 65.5·44-s + ⋯ |
| L(s) = 1 | − 0.158·2-s − 0.974·4-s − 0.172·5-s − 0.439·7-s + 0.312·8-s + 0.0273·10-s − 0.230·11-s + 0.0695·14-s + 0.925·16-s + 0.743·17-s + 0.589·19-s + 0.168·20-s + 0.0364·22-s − 0.806·23-s − 0.970·25-s + 0.428·28-s − 1.22·29-s + 0.667·31-s − 0.458·32-s − 0.117·34-s + 0.0759·35-s − 0.607·37-s − 0.0932·38-s − 0.0539·40-s − 1.66·41-s + 0.716·43-s + 0.224·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8473317377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8473317377\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.447T + 8T^{2} \) |
| 5 | \( 1 + 1.93T + 125T^{2} \) |
| 7 | \( 1 + 8.14T + 343T^{2} \) |
| 11 | \( 1 + 8.40T + 1.33e3T^{2} \) |
| 17 | \( 1 - 52.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 48.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 88.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 136.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 436.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 202.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 618.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 453.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 500.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 886.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 123.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 673.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 681.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 939.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 754.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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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
−9.228542879295009268263006029405, −8.182686296944160117321659309202, −7.80130408552756511722725777750, −6.69814660612442805374621825323, −5.65716400472126640485810735540, −5.02006157556311671230808507964, −3.89224824965135471437911832982, −3.29226130163077883311923699503, −1.77511622316004472910606236005, −0.46277512661319383567529125119,
0.46277512661319383567529125119, 1.77511622316004472910606236005, 3.29226130163077883311923699503, 3.89224824965135471437911832982, 5.02006157556311671230808507964, 5.65716400472126640485810735540, 6.69814660612442805374621825323, 7.80130408552756511722725777750, 8.182686296944160117321659309202, 9.228542879295009268263006029405
