| L(s) = 1 | + 5·5-s + 12.8·7-s − 18.2·11-s + 20.2·13-s + 6.65·17-s − 150.·19-s + 88.1·23-s + 25·25-s − 201.·29-s + 268.·31-s + 64.3·35-s − 123.·37-s + 275.·41-s − 488.·43-s − 436.·47-s − 177.·49-s + 340.·53-s − 91.2·55-s − 548.·59-s − 206.·61-s + 101.·65-s − 499.·67-s + 460.·71-s − 416.·73-s − 234.·77-s + 289.·79-s − 909.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.694·7-s − 0.500·11-s + 0.432·13-s + 0.0949·17-s − 1.82·19-s + 0.798·23-s + 0.200·25-s − 1.29·29-s + 1.55·31-s + 0.310·35-s − 0.548·37-s + 1.04·41-s − 1.73·43-s − 1.35·47-s − 0.517·49-s + 0.882·53-s − 0.223·55-s − 1.21·59-s − 0.434·61-s + 0.193·65-s − 0.911·67-s + 0.770·71-s − 0.668·73-s − 0.347·77-s + 0.412·79-s − 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 12.8T + 343T^{2} \) |
| 11 | \( 1 + 18.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.65T + 4.91e3T^{2} \) |
| 19 | \( 1 + 150.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 88.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 201.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 268.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 123.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 275.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 488.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 436.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 340.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 548.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 206.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 499.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 460.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 416.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 289.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 909.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 186.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 648.T + 9.12e5T^{2} \) |
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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
−8.387526405487466018358660875033, −7.67682805048906370127508881385, −6.66004149579822095053248657075, −6.02843975436681544320287897534, −5.05235395478381034850180219421, −4.43266241555921281444754296037, −3.28309569594611300921282391616, −2.22816492338724674413679441763, −1.39349413646942090878245744678, 0,
1.39349413646942090878245744678, 2.22816492338724674413679441763, 3.28309569594611300921282391616, 4.43266241555921281444754296037, 5.05235395478381034850180219421, 6.02843975436681544320287897534, 6.66004149579822095053248657075, 7.67682805048906370127508881385, 8.387526405487466018358660875033
