| L(s) = 1 | − 1.12i·2-s − i·3-s + 0.740·4-s − 1.12·6-s + 1.11i·7-s − 3.07i·8-s − 9-s + 3.67·11-s − 0.740i·12-s + 4.05i·13-s + 1.24·14-s − 1.97·16-s + 2.12i·17-s + 1.12i·18-s + 4.06·19-s + ⋯ |
| L(s) = 1 | − 0.793i·2-s − 0.577i·3-s + 0.370·4-s − 0.458·6-s + 0.420i·7-s − 1.08i·8-s − 0.333·9-s + 1.10·11-s − 0.213i·12-s + 1.12i·13-s + 0.334·14-s − 0.492·16-s + 0.514i·17-s + 0.264i·18-s + 0.931·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.281772399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.281772399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.12iT - 2T^{2} \) |
| 7 | \( 1 - 1.11iT - 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 13 | \( 1 - 4.05iT - 13T^{2} \) |
| 17 | \( 1 - 2.12iT - 17T^{2} \) |
| 19 | \( 1 - 4.06T + 19T^{2} \) |
| 23 | \( 1 + 6.17iT - 23T^{2} \) |
| 29 | \( 1 - 2.25T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 7.37iT - 37T^{2} \) |
| 41 | \( 1 + 7.47T + 41T^{2} \) |
| 43 | \( 1 + 9.24iT - 43T^{2} \) |
| 47 | \( 1 + 3.12iT - 47T^{2} \) |
| 53 | \( 1 - 3.50iT - 53T^{2} \) |
| 59 | \( 1 - 6.59T + 59T^{2} \) |
| 61 | \( 1 + 9.10T + 61T^{2} \) |
| 67 | \( 1 - 2.62iT - 67T^{2} \) |
| 71 | \( 1 - 0.660T + 71T^{2} \) |
| 73 | \( 1 + 7.47iT - 73T^{2} \) |
| 79 | \( 1 + 8.53T + 79T^{2} \) |
| 83 | \( 1 + 12.2iT - 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 13.4iT - 97T^{2} \) |
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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
−9.018876251202670717571568706681, −8.453855334425782718008042677770, −7.29529416990370875756492811034, −6.55901304891817185839011005343, −6.19694081191112642061867495281, −4.77371039492208758844182892344, −3.82655868048389351555315284703, −2.84012827946015553633581565282, −1.92921459283886267641149793451, −1.03642240310726385296840065567,
1.16851267723314955359698737618, 2.75578290418188100042169799263, 3.59064361697748101519621315424, 4.72619640151760915191788394814, 5.50501356339353380412457561098, 6.24196482264621041243310897715, 7.08968853943496045303895618037, 7.75650507780325950243989268057, 8.482359357450302636715357332571, 9.436006095706176327894675338217
