| L(s) = 1 | + 2i·3-s + (−5 − 10i)5-s − 26i·7-s + 23·9-s + 28·11-s − 12i·13-s + (20 − 10i)15-s − 64i·17-s − 60·19-s + 52·21-s − 58i·23-s + (−75 + 100i)25-s + 100i·27-s − 90·29-s + 128·31-s + ⋯ |
| L(s) = 1 | + 0.384i·3-s + (−0.447 − 0.894i)5-s − 1.40i·7-s + 0.851·9-s + 0.767·11-s − 0.256i·13-s + (0.344 − 0.172i)15-s − 0.913i·17-s − 0.724·19-s + 0.540·21-s − 0.525i·23-s + (−0.599 + 0.800i)25-s + 0.712i·27-s − 0.576·29-s + 0.741·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.18198 - 0.730504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18198 - 0.730504i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (5 + 10i)T \) |
| good | 3 | \( 1 - 2iT - 27T^{2} \) |
| 7 | \( 1 + 26iT - 343T^{2} \) |
| 11 | \( 1 - 28T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 64iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 60T + 6.85e3T^{2} \) |
| 23 | \( 1 + 58iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 90T + 2.43e4T^{2} \) |
| 31 | \( 1 - 128T + 2.97e4T^{2} \) |
| 37 | \( 1 - 236iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 242T + 6.89e4T^{2} \) |
| 43 | \( 1 - 362iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 226iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 108iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 20T + 2.05e5T^{2} \) |
| 61 | \( 1 - 542T + 2.26e5T^{2} \) |
| 67 | \( 1 - 434iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 632iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 720T + 4.93e5T^{2} \) |
| 83 | \( 1 + 478iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 490T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.45e3iT - 9.12e5T^{2} \) |
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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
−13.58321738757912432576907864682, −12.73159405039226637921851587056, −11.48707051583408906738104714388, −10.30645565126906576086136323070, −9.313257126789916623075637991462, −7.920207403941555692963667016788, −6.78583866459021974187339627800, −4.73074108874878233178853455269, −3.91516621417826761121525157483, −0.957836559168743372744539639026,
2.13212273937316211592754290477, 3.95713950107976570085892972117, 5.97011712467024798075455849164, 7.00360623711879877267743546589, 8.330440342658319081820298780338, 9.581523675084277520018602528974, 10.94615263446431194212458318343, 12.00601385006042567280648410426, 12.78109113831846301638774414288, 14.24871770551616562608790852176
