| L(s) = 1 | − 1.48·2-s + 0.193·4-s − 5-s − 3.28·7-s + 2.67·8-s + 1.48·10-s + 2.86·11-s − 7.02·13-s + 4.86·14-s − 4.35·16-s + 1.84·17-s + 19-s − 0.193·20-s − 4.24·22-s − 0.156·23-s + 25-s + 10.4·26-s − 0.637·28-s − 9.18·29-s + 9.50·31-s + 1.09·32-s − 2.73·34-s + 3.28·35-s − 1.36·37-s − 1.48·38-s − 2.67·40-s + 9.05·41-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.0969·4-s − 0.447·5-s − 1.24·7-s + 0.945·8-s + 0.468·10-s + 0.865·11-s − 1.94·13-s + 1.30·14-s − 1.08·16-s + 0.447·17-s + 0.229·19-s − 0.0433·20-s − 0.906·22-s − 0.0325·23-s + 0.200·25-s + 2.04·26-s − 0.120·28-s − 1.70·29-s + 1.70·31-s + 0.193·32-s − 0.468·34-s + 0.555·35-s − 0.223·37-s − 0.240·38-s − 0.422·40-s + 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5059950071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5059950071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 13 | \( 1 + 7.02T + 13T^{2} \) |
| 17 | \( 1 - 1.84T + 17T^{2} \) |
| 23 | \( 1 + 0.156T + 23T^{2} \) |
| 29 | \( 1 + 9.18T + 29T^{2} \) |
| 31 | \( 1 - 9.50T + 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 - 9.05T + 41T^{2} \) |
| 43 | \( 1 - 5.59T + 43T^{2} \) |
| 47 | \( 1 - 4.93T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 1.35T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 6.96T + 71T^{2} \) |
| 73 | \( 1 - 2.57T + 73T^{2} \) |
| 79 | \( 1 - 8.18T + 79T^{2} \) |
| 83 | \( 1 + 8.46T + 83T^{2} \) |
| 89 | \( 1 - 8.16T + 89T^{2} \) |
| 97 | \( 1 - 7.98T + 97T^{2} \) |
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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
−9.782824091161358929434273326621, −9.533182216748146724307524685850, −8.698823896756403412852180295935, −7.48190064251216947421290971650, −7.22769828669111755018584315854, −6.00890703600252494531148289968, −4.72593415863024615621178334965, −3.75112131034574924967399660589, −2.43829963064028097798459743381, −0.65463117153190541741788343508,
0.65463117153190541741788343508, 2.43829963064028097798459743381, 3.75112131034574924967399660589, 4.72593415863024615621178334965, 6.00890703600252494531148289968, 7.22769828669111755018584315854, 7.48190064251216947421290971650, 8.698823896756403412852180295935, 9.533182216748146724307524685850, 9.782824091161358929434273326621
