| L(s) = 1 | − 0.167·2-s − 1.97·4-s + 2.16·5-s + 3.80·7-s + 0.665·8-s − 0.362·10-s + 11-s − 0.167·13-s − 0.637·14-s + 3.83·16-s − 1.13·17-s − 4.63·19-s − 4.27·20-s − 0.167·22-s + 4.13·23-s − 0.302·25-s + 0.0280·26-s − 7.50·28-s + 9.97·29-s − 1.02·31-s − 1.97·32-s + 0.190·34-s + 8.24·35-s − 4.94·37-s + 0.776·38-s + 1.44·40-s − 2.97·41-s + ⋯ |
| L(s) = 1 | − 0.118·2-s − 0.985·4-s + 0.969·5-s + 1.43·7-s + 0.235·8-s − 0.114·10-s + 0.301·11-s − 0.0464·13-s − 0.170·14-s + 0.958·16-s − 0.276·17-s − 1.06·19-s − 0.955·20-s − 0.0357·22-s + 0.863·23-s − 0.0604·25-s + 0.00549·26-s − 1.41·28-s + 1.85·29-s − 0.184·31-s − 0.348·32-s + 0.0327·34-s + 1.39·35-s − 0.812·37-s + 0.125·38-s + 0.227·40-s − 0.464·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.670987905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.670987905\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.167T + 2T^{2} \) |
| 5 | \( 1 - 2.16T + 5T^{2} \) |
| 7 | \( 1 - 3.80T + 7T^{2} \) |
| 13 | \( 1 + 0.167T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 + 4.63T + 19T^{2} \) |
| 23 | \( 1 - 4.13T + 23T^{2} \) |
| 29 | \( 1 - 9.97T + 29T^{2} \) |
| 31 | \( 1 + 1.02T + 31T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 + 2.97T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 - 6.63T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 2.80T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 7.44T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 8.02T + 89T^{2} \) |
| 97 | \( 1 - 6.27T + 97T^{2} \) |
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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
−10.23551248554371332264292310984, −9.034659748140168951726846094451, −8.697287105503682778645045988066, −7.80013618321647868680495718775, −6.64549732038620289286041608268, −5.54779331247304747450148310924, −4.85238249206593321434519936238, −4.04088199767727839339744429041, −2.35353106777765548274119533919, −1.18072091599964512308474894277,
1.18072091599964512308474894277, 2.35353106777765548274119533919, 4.04088199767727839339744429041, 4.85238249206593321434519936238, 5.54779331247304747450148310924, 6.64549732038620289286041608268, 7.80013618321647868680495718775, 8.697287105503682778645045988066, 9.034659748140168951726846094451, 10.23551248554371332264292310984
