| L(s) = 1 | − 2.48·2-s + 3-s + 4.15·4-s − 0.0418·5-s − 2.48·6-s − 4.14·7-s − 5.34·8-s + 9-s + 0.103·10-s − 11-s + 4.15·12-s + 10.2·14-s − 0.0418·15-s + 4.94·16-s − 4.04·17-s − 2.48·18-s − 1.07·19-s − 0.173·20-s − 4.14·21-s + 2.48·22-s + 9.21·23-s − 5.34·24-s − 4.99·25-s + 27-s − 17.2·28-s + 0.454·29-s + 0.103·30-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 0.577·3-s + 2.07·4-s − 0.0187·5-s − 1.01·6-s − 1.56·7-s − 1.88·8-s + 0.333·9-s + 0.0328·10-s − 0.301·11-s + 1.19·12-s + 2.75·14-s − 0.0108·15-s + 1.23·16-s − 0.980·17-s − 0.584·18-s − 0.247·19-s − 0.0388·20-s − 0.905·21-s + 0.528·22-s + 1.92·23-s − 1.09·24-s − 0.999·25-s + 0.192·27-s − 3.25·28-s + 0.0843·29-s + 0.0189·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.48T + 2T^{2} \) |
| 5 | \( 1 + 0.0418T + 5T^{2} \) |
| 7 | \( 1 + 4.14T + 7T^{2} \) |
| 17 | \( 1 + 4.04T + 17T^{2} \) |
| 19 | \( 1 + 1.07T + 19T^{2} \) |
| 23 | \( 1 - 9.21T + 23T^{2} \) |
| 29 | \( 1 - 0.454T + 29T^{2} \) |
| 31 | \( 1 - 4.20T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 4.04T + 41T^{2} \) |
| 43 | \( 1 - 6.63T + 43T^{2} \) |
| 47 | \( 1 + 5.44T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 - 8.93T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 5.35T + 71T^{2} \) |
| 73 | \( 1 + 9.96T + 73T^{2} \) |
| 79 | \( 1 - 4.52T + 79T^{2} \) |
| 83 | \( 1 - 7.60T + 83T^{2} \) |
| 89 | \( 1 - 2.71T + 89T^{2} \) |
| 97 | \( 1 - 8.46T + 97T^{2} \) |
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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
−7.917410086121950316236572144247, −7.30397016274225808535090833846, −6.50933224730884900399791095417, −6.28490031469609099742628536296, −4.88134146876106777442600782342, −3.73341073002807255826141680760, −2.82059232136905499031918425918, −2.34181339713749883930989112872, −1.04892384781272638987109216838, 0,
1.04892384781272638987109216838, 2.34181339713749883930989112872, 2.82059232136905499031918425918, 3.73341073002807255826141680760, 4.88134146876106777442600782342, 6.28490031469609099742628536296, 6.50933224730884900399791095417, 7.30397016274225808535090833846, 7.917410086121950316236572144247
