| L(s) = 1 | + 2.73·2-s + 3-s + 5.50·4-s − 2.84·5-s + 2.73·6-s + 3.93·7-s + 9.58·8-s + 9-s − 7.78·10-s + 11-s + 5.50·12-s + 10.7·14-s − 2.84·15-s + 15.2·16-s + 3.81·17-s + 2.73·18-s − 2.94·19-s − 15.6·20-s + 3.93·21-s + 2.73·22-s − 1.89·23-s + 9.58·24-s + 3.07·25-s + 27-s + 21.6·28-s − 2.09·29-s − 7.78·30-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 0.577·3-s + 2.75·4-s − 1.27·5-s + 1.11·6-s + 1.48·7-s + 3.38·8-s + 0.333·9-s − 2.46·10-s + 0.301·11-s + 1.58·12-s + 2.87·14-s − 0.733·15-s + 3.81·16-s + 0.926·17-s + 0.645·18-s − 0.674·19-s − 3.49·20-s + 0.857·21-s + 0.583·22-s − 0.395·23-s + 1.95·24-s + 0.614·25-s + 0.192·27-s + 4.08·28-s − 0.389·29-s − 1.42·30-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
| \(L(1)\) |
\(\approx\) |
\(9.211795358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.211795358\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.73T + 2T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 17 | \( 1 - 3.81T + 17T^{2} \) |
| 19 | \( 1 + 2.94T + 19T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 + 2.09T + 29T^{2} \) |
| 31 | \( 1 + 6.16T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 - 6.35T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 5.31T + 47T^{2} \) |
| 53 | \( 1 + 2.37T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 + 2.34T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 1.57T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 3.23T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−7.892838682134381843111245174970, −7.45454971939032172284494616512, −6.66911378224159007786151717799, −5.70791805263340230571278081398, −5.04126758109308459294655539580, −4.24339052114726560853597812940, −3.97149821567742913286491795951, −3.16200787216044213230439990459, −2.20191019366229916208339818894, −1.36815985258767532950655311011,
1.36815985258767532950655311011, 2.20191019366229916208339818894, 3.16200787216044213230439990459, 3.97149821567742913286491795951, 4.24339052114726560853597812940, 5.04126758109308459294655539580, 5.70791805263340230571278081398, 6.66911378224159007786151717799, 7.45454971939032172284494616512, 7.892838682134381843111245174970
