| L(s) = 1 | + (0.911 − 1.47i)3-s − 1.56i·5-s − 5.19i·7-s + (−1.33 − 2.68i)9-s + 3.94·11-s − 4.57·13-s + (−2.29 − 1.42i)15-s + 0.943i·17-s + i·19-s + (−7.65 − 4.73i)21-s − 0.478·23-s + 2.56·25-s + (−5.17 − 0.476i)27-s − 9.53i·29-s + 2.67i·31-s + ⋯ |
| L(s) = 1 | + (0.526 − 0.850i)3-s − 0.697i·5-s − 1.96i·7-s + (−0.446 − 0.895i)9-s + 1.19·11-s − 1.26·13-s + (−0.593 − 0.367i)15-s + 0.228i·17-s + 0.229i·19-s + (−1.67 − 1.03i)21-s − 0.0997·23-s + 0.512·25-s + (−0.995 − 0.0917i)27-s − 1.77i·29-s + 0.480i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.874687758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.874687758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.911 + 1.47i)T \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 + 1.56iT - 5T^{2} \) |
| 7 | \( 1 + 5.19iT - 7T^{2} \) |
| 11 | \( 1 - 3.94T + 11T^{2} \) |
| 13 | \( 1 + 4.57T + 13T^{2} \) |
| 17 | \( 1 - 0.943iT - 17T^{2} \) |
| 23 | \( 1 + 0.478T + 23T^{2} \) |
| 29 | \( 1 + 9.53iT - 29T^{2} \) |
| 31 | \( 1 - 2.67iT - 31T^{2} \) |
| 37 | \( 1 - 6.67T + 37T^{2} \) |
| 41 | \( 1 - 0.718iT - 41T^{2} \) |
| 43 | \( 1 - 4.49iT - 43T^{2} \) |
| 47 | \( 1 - 8.30T + 47T^{2} \) |
| 53 | \( 1 - 13.4iT - 53T^{2} \) |
| 59 | \( 1 - 6.71T + 59T^{2} \) |
| 61 | \( 1 - 9.09T + 61T^{2} \) |
| 67 | \( 1 - 11.9iT - 67T^{2} \) |
| 71 | \( 1 + 16.1T + 71T^{2} \) |
| 73 | \( 1 + 2.58T + 73T^{2} \) |
| 79 | \( 1 - 9.22iT - 79T^{2} \) |
| 83 | \( 1 - 4.07T + 83T^{2} \) |
| 89 | \( 1 + 15.6iT - 89T^{2} \) |
| 97 | \( 1 - 1.01T + 97T^{2} \) |
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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
−8.827777021364488179934572340624, −7.944977756311640253824465981359, −7.33965337828159534556429614015, −6.79279721597564142568415067260, −5.85955954692769550235715742358, −4.36595938451617929930334222303, −4.10863568438391838866616536732, −2.79263057489982108827190730138, −1.42574860642346083778916366659, −0.67042180698146173640564592024,
2.12782422010950138034653446977, 2.74124796532993180354689824794, 3.61760842259177864309114081749, 4.83668077142588218500572274118, 5.42329461111174532376973033542, 6.41247764533774400036774333540, 7.25498570123394476279125968069, 8.322156083308038246442427914945, 9.085559799347125522798579205260, 9.354815938486468231338835231933
