| L(s) = 1 | + 2.54·2-s − 2.15·3-s + 4.48·4-s − 5.48·6-s + 2.93·7-s + 6.31·8-s + 1.63·9-s + 0.635·11-s − 9.64·12-s + 7.48·14-s + 7.11·16-s − 1.22·17-s + 4.16·18-s − 1.36·19-s − 6.32·21-s + 1.61·22-s + 2.15·23-s − 13.5·24-s + 2.93·27-s + 13.1·28-s + 3·29-s + 8.96·31-s + 5.48·32-s − 1.36·33-s − 3.11·34-s + 7.32·36-s − 1.22·37-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 1.24·3-s + 2.24·4-s − 2.23·6-s + 1.11·7-s + 2.23·8-s + 0.545·9-s + 0.191·11-s − 2.78·12-s + 1.99·14-s + 1.77·16-s − 0.296·17-s + 0.981·18-s − 0.313·19-s − 1.38·21-s + 0.344·22-s + 0.448·23-s − 2.77·24-s + 0.565·27-s + 2.48·28-s + 0.557·29-s + 1.60·31-s + 0.969·32-s − 0.238·33-s − 0.534·34-s + 1.22·36-s − 0.201·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.661274604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.661274604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.54T + 2T^{2} \) |
| 3 | \( 1 + 2.15T + 3T^{2} \) |
| 7 | \( 1 - 2.93T + 7T^{2} \) |
| 11 | \( 1 - 0.635T + 11T^{2} \) |
| 17 | \( 1 + 1.22T + 17T^{2} \) |
| 19 | \( 1 + 1.36T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 8.96T + 31T^{2} \) |
| 37 | \( 1 + 1.22T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 + 1.36T + 43T^{2} \) |
| 47 | \( 1 - 6.16T + 47T^{2} \) |
| 53 | \( 1 - 0.642T + 53T^{2} \) |
| 59 | \( 1 + 7.59T + 59T^{2} \) |
| 61 | \( 1 + 2.27T + 61T^{2} \) |
| 67 | \( 1 + 8.03T + 67T^{2} \) |
| 71 | \( 1 + 2.63T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 1.03T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.091500956271396683948654993177, −7.28984581365240312883331863249, −6.48097332958984943574183709407, −6.03546076134520900756507027728, −5.31394101547657503750043169911, −4.60335289905166966189974170166, −4.37802272702170716333166625957, −3.12194422943326834700673441747, −2.19597606349812059824651486196, −1.04774340187716176766708974778,
1.04774340187716176766708974778, 2.19597606349812059824651486196, 3.12194422943326834700673441747, 4.37802272702170716333166625957, 4.60335289905166966189974170166, 5.31394101547657503750043169911, 6.03546076134520900756507027728, 6.48097332958984943574183709407, 7.28984581365240312883331863249, 8.091500956271396683948654993177
