| L(s) = 1 | + 0.381i·2-s + i·3-s + 1.85·4-s − 0.381·6-s + 3.61i·7-s + 1.47i·8-s − 9-s − 1.76·11-s + 1.85i·12-s − 3i·13-s − 1.38·14-s + 3.14·16-s + 5.61i·17-s − 0.381i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 0.270i·2-s + 0.577i·3-s + 0.927·4-s − 0.155·6-s + 1.36i·7-s + 0.520i·8-s − 0.333·9-s − 0.531·11-s + 0.535i·12-s − 0.832i·13-s − 0.369·14-s + 0.786·16-s + 1.36i·17-s − 0.0900i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12352 + 1.12352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12352 + 1.12352i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.381iT - 2T^{2} \) |
| 7 | \( 1 - 3.61iT - 7T^{2} \) |
| 11 | \( 1 + 1.76T + 11T^{2} \) |
| 13 | \( 1 + 3iT - 13T^{2} \) |
| 17 | \( 1 - 5.61iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 + 6.70iT - 23T^{2} \) |
| 29 | \( 1 + 0.236T + 29T^{2} \) |
| 31 | \( 1 - 8.09T + 31T^{2} \) |
| 37 | \( 1 - 5iT - 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 4.61iT - 43T^{2} \) |
| 47 | \( 1 + 13.1iT - 47T^{2} \) |
| 53 | \( 1 - 1.38iT - 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 - 6.09T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 7.85T + 71T^{2} \) |
| 73 | \( 1 + 15.8iT - 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 + 7.32iT - 83T^{2} \) |
| 89 | \( 1 + 3.47T + 89T^{2} \) |
| 97 | \( 1 + 10.5iT - 97T^{2} \) |
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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
−11.66323550430191610665131831883, −10.54186932463729220909735997930, −10.04710405486447815158107420346, −8.450600351578932586482631046189, −8.249218974456389072503703849526, −6.64298613900243841340829774688, −5.83702680614575928148881721426, −4.98426828630885020189546190149, −3.23434246316775919176751009167, −2.23511147531799918772124412750,
1.14484904531939549970433299779, 2.60651805165480462728901073425, 3.87074953732194923329507354264, 5.34436098192352677636921996920, 6.78093385471798345643467189250, 7.17129867272584766659044160619, 8.048992203311119554993948482167, 9.572992759460655469016312435682, 10.33335557779831771509534690818, 11.39359785704980625515268897074
