| L(s) = 1 | − 2-s + 4-s − 2.03·5-s − 4.04·7-s − 8-s + 2.03·10-s + 3.73·11-s + 6.11·13-s + 4.04·14-s + 16-s + 0.0115·17-s − 7.09·19-s − 2.03·20-s − 3.73·22-s + 1.44·23-s − 0.856·25-s − 6.11·26-s − 4.04·28-s + 8.03·29-s − 8.40·31-s − 32-s − 0.0115·34-s + 8.22·35-s + 3.14·37-s + 7.09·38-s + 2.03·40-s + 0.612·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.910·5-s − 1.52·7-s − 0.353·8-s + 0.643·10-s + 1.12·11-s + 1.69·13-s + 1.07·14-s + 0.250·16-s + 0.00280·17-s − 1.62·19-s − 0.455·20-s − 0.796·22-s + 0.301·23-s − 0.171·25-s − 1.19·26-s − 0.763·28-s + 1.49·29-s − 1.50·31-s − 0.176·32-s − 0.00198·34-s + 1.39·35-s + 0.517·37-s + 1.15·38-s + 0.321·40-s + 0.0955·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7520135517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7520135517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 + 2.03T + 5T^{2} \) |
| 7 | \( 1 + 4.04T + 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 - 6.11T + 13T^{2} \) |
| 17 | \( 1 - 0.0115T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 - 1.44T + 23T^{2} \) |
| 29 | \( 1 - 8.03T + 29T^{2} \) |
| 31 | \( 1 + 8.40T + 31T^{2} \) |
| 37 | \( 1 - 3.14T + 37T^{2} \) |
| 41 | \( 1 - 0.612T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 5.57T + 47T^{2} \) |
| 53 | \( 1 - 4.78T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 8.80T + 61T^{2} \) |
| 67 | \( 1 + 7.00T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 1.40T + 73T^{2} \) |
| 79 | \( 1 - 9.90T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 8.69T + 97T^{2} \) |
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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
−8.590949470248775631185390028458, −7.87886267076471531332640841363, −6.73697517606427235589018661453, −6.56294207618252276975868666066, −5.87708007262833629687109522901, −4.36917913958927988396600999134, −3.66400136893904226729703330665, −3.16272706422288923827289384548, −1.75631696328033989414087816250, −0.55127317512670912999159688718,
0.55127317512670912999159688718, 1.75631696328033989414087816250, 3.16272706422288923827289384548, 3.66400136893904226729703330665, 4.36917913958927988396600999134, 5.87708007262833629687109522901, 6.56294207618252276975868666066, 6.73697517606427235589018661453, 7.87886267076471531332640841363, 8.590949470248775631185390028458
