| L(s) = 1 | + 0.612·5-s + 22.7·7-s + 60.2·11-s − 52.9·13-s − 47.1·17-s − 29.1·19-s + 109.·23-s − 124.·25-s − 10.4·29-s − 220.·31-s + 13.9·35-s − 408.·37-s − 360.·41-s + 236.·43-s − 129.·47-s + 174.·49-s + 117.·53-s + 36.9·55-s − 262.·59-s − 273.·61-s − 32.4·65-s − 89.4·67-s − 350.·71-s − 532.·73-s + 1.37e3·77-s − 166.·79-s − 361.·83-s + ⋯ |
| L(s) = 1 | + 0.0547·5-s + 1.22·7-s + 1.65·11-s − 1.12·13-s − 0.672·17-s − 0.352·19-s + 0.992·23-s − 0.996·25-s − 0.0667·29-s − 1.27·31-s + 0.0672·35-s − 1.81·37-s − 1.37·41-s + 0.838·43-s − 0.400·47-s + 0.508·49-s + 0.305·53-s + 0.0905·55-s − 0.580·59-s − 0.573·61-s − 0.0618·65-s − 0.163·67-s − 0.585·71-s − 0.853·73-s + 2.02·77-s − 0.237·79-s − 0.478·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 0.612T + 125T^{2} \) |
| 7 | \( 1 - 22.7T + 343T^{2} \) |
| 11 | \( 1 - 60.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 29.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 109.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 10.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 220.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 408.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 360.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 236.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 117.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 262.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 273.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 89.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 350.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 532.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 166.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 361.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 40.3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 614.T + 9.12e5T^{2} \) |
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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
−8.413465112343356700181975840405, −7.32621853969158218206902649164, −6.93869053722156150634991688501, −5.87110118822486464694552313330, −4.96258133061725132884736292131, −4.34810298678029247321739979694, −3.40824580271926980109637073808, −2.03574209211811604778092805490, −1.46767376751878715206919523584, 0,
1.46767376751878715206919523584, 2.03574209211811604778092805490, 3.40824580271926980109637073808, 4.34810298678029247321739979694, 4.96258133061725132884736292131, 5.87110118822486464694552313330, 6.93869053722156150634991688501, 7.32621853969158218206902649164, 8.413465112343356700181975840405
