| L(s) = 1 | + 3·3-s − 2.70·5-s + 9·9-s − 64.1·11-s + 70.8·13-s − 8.11·15-s − 14.7·17-s − 159.·19-s + 172.·23-s − 117.·25-s + 27·27-s + 18.2·29-s − 149.·31-s − 192.·33-s + 41.7·37-s + 212.·39-s − 50.2·41-s + 388·43-s − 24.3·45-s + 494.·47-s − 44.1·51-s + 469.·53-s + 173.·55-s − 477.·57-s − 343.·59-s − 72.4·61-s − 191.·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.241·5-s + 0.333·9-s − 1.75·11-s + 1.51·13-s − 0.139·15-s − 0.209·17-s − 1.92·19-s + 1.56·23-s − 0.941·25-s + 0.192·27-s + 0.116·29-s − 0.865·31-s − 1.01·33-s + 0.185·37-s + 0.872·39-s − 0.191·41-s + 1.37·43-s − 0.0806·45-s + 1.53·47-s − 0.121·51-s + 1.21·53-s + 0.425·55-s − 1.10·57-s − 0.757·59-s − 0.152·61-s − 0.365·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.108319725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.108319725\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2.70T + 125T^{2} \) |
| 11 | \( 1 + 64.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 14.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 159.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 172.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 18.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 149.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 41.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 50.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 388T + 7.95e4T^{2} \) |
| 47 | \( 1 - 494.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 469.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 343.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 72.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 293.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 629.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 696.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 640.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 667.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.30e3T + 9.12e5T^{2} \) |
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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
−8.738318990353735905656298600727, −7.890041417232435301536605048108, −7.31418063749322186238240207856, −6.27277263517035863369481384508, −5.53851373381353917707128931939, −4.50015044359038879456892995088, −3.74744236762429062186722687686, −2.77009233555504317668401669513, −1.97624827628271108411789854150, −0.60978641799910065497085969334,
0.60978641799910065497085969334, 1.97624827628271108411789854150, 2.77009233555504317668401669513, 3.74744236762429062186722687686, 4.50015044359038879456892995088, 5.53851373381353917707128931939, 6.27277263517035863369481384508, 7.31418063749322186238240207856, 7.890041417232435301536605048108, 8.738318990353735905656298600727
