| L(s) = 1 | − 8.05·3-s − 11.6·5-s + 24.4·7-s + 37.8·9-s − 18.4·11-s − 51.7·13-s + 93.8·15-s − 10.4·17-s + 22.3·19-s − 196.·21-s − 23·23-s + 10.6·25-s − 87.5·27-s + 167.·29-s − 176.·31-s + 148.·33-s − 284.·35-s + 35.8·37-s + 416.·39-s + 412.·41-s + 192.·43-s − 441.·45-s − 591.·47-s + 253.·49-s + 84.2·51-s + 750.·53-s + 214.·55-s + ⋯ |
| L(s) = 1 | − 1.54·3-s − 1.04·5-s + 1.31·7-s + 1.40·9-s − 0.504·11-s − 1.10·13-s + 1.61·15-s − 0.149·17-s + 0.269·19-s − 2.04·21-s − 0.208·23-s + 0.0851·25-s − 0.623·27-s + 1.07·29-s − 1.02·31-s + 0.782·33-s − 1.37·35-s + 0.159·37-s + 1.70·39-s + 1.57·41-s + 0.680·43-s − 1.46·45-s − 1.83·47-s + 0.738·49-s + 0.231·51-s + 1.94·53-s + 0.525·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 8.05T + 27T^{2} \) |
| 5 | \( 1 + 11.6T + 125T^{2} \) |
| 7 | \( 1 - 24.4T + 343T^{2} \) |
| 11 | \( 1 + 18.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 51.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 22.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 167.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 176.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 35.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 412.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 192.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 591.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 750.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 198.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 264.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 155.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.08e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 224.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 261.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 15.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 200.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 113.T + 9.12e5T^{2} \) |
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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
−8.534711307546955134019378937796, −7.69207956088593966705237492134, −7.27055227026367480554876731187, −6.17460414346098430825799183136, −5.16146017559385351298653132499, −4.81778249355547888181889477750, −3.92394145637932955475008160990, −2.33256478868911951244009084323, −0.954681742448365421501653938920, 0,
0.954681742448365421501653938920, 2.33256478868911951244009084323, 3.92394145637932955475008160990, 4.81778249355547888181889477750, 5.16146017559385351298653132499, 6.17460414346098430825799183136, 7.27055227026367480554876731187, 7.69207956088593966705237492134, 8.534711307546955134019378937796
