| L(s) = 1 | + 2.89·3-s − 6.53·5-s + 24.8·7-s − 18.5·9-s + 72.3·13-s − 18.9·15-s + 5.55·17-s − 39.2·19-s + 71.9·21-s + 172.·23-s − 82.3·25-s − 132.·27-s − 13.5·29-s − 105.·31-s − 162.·35-s + 435.·37-s + 209.·39-s − 65.7·41-s − 230.·43-s + 121.·45-s + 245.·47-s + 272.·49-s + 16.1·51-s + 300.·53-s − 113.·57-s − 87.9·59-s + 333.·61-s + ⋯ |
| L(s) = 1 | + 0.557·3-s − 0.584·5-s + 1.33·7-s − 0.688·9-s + 1.54·13-s − 0.325·15-s + 0.0792·17-s − 0.473·19-s + 0.747·21-s + 1.55·23-s − 0.658·25-s − 0.942·27-s − 0.0870·29-s − 0.609·31-s − 0.782·35-s + 1.93·37-s + 0.860·39-s − 0.250·41-s − 0.818·43-s + 0.402·45-s + 0.761·47-s + 0.795·49-s + 0.0442·51-s + 0.777·53-s − 0.264·57-s − 0.194·59-s + 0.700·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.743352847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.743352847\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.89T + 27T^{2} \) |
| 5 | \( 1 + 6.53T + 125T^{2} \) |
| 7 | \( 1 - 24.8T + 343T^{2} \) |
| 13 | \( 1 - 72.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.55T + 4.91e3T^{2} \) |
| 19 | \( 1 + 39.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 13.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 105.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 435.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 65.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 230.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 245.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 87.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 333.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 345.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 656.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.06e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 535.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 31.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 522.T + 9.12e5T^{2} \) |
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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
−9.367766038139586665489657603866, −8.560208952503289567966193744726, −8.177173854270469218992078971226, −7.37099716749062776489807065578, −6.15627296664256118788647731284, −5.22714416371591242870777939172, −4.17910558087765213837533230803, −3.33857752837111428839886159112, −2.10585275306934672073439911399, −0.900889156645437734143864796719,
0.900889156645437734143864796719, 2.10585275306934672073439911399, 3.33857752837111428839886159112, 4.17910558087765213837533230803, 5.22714416371591242870777939172, 6.15627296664256118788647731284, 7.37099716749062776489807065578, 8.177173854270469218992078971226, 8.560208952503289567966193744726, 9.367766038139586665489657603866
