| L(s) = 1 | + (1.14 − 1.30i)3-s − 5-s + 5.02i·7-s + (−0.396 − 2.97i)9-s − 4.49·11-s + 4.52i·13-s + (−1.14 + 1.30i)15-s + 1.07i·17-s + 2.27·19-s + (6.54 + 5.73i)21-s − 8.57i·23-s + 25-s + (−4.32 − 2.87i)27-s − 1.56i·29-s − 8.54i·31-s + ⋯ |
| L(s) = 1 | + (0.658 − 0.752i)3-s − 0.447·5-s + 1.89i·7-s + (−0.132 − 0.991i)9-s − 1.35·11-s + 1.25i·13-s + (−0.294 + 0.336i)15-s + 0.261i·17-s + 0.522·19-s + (1.42 + 1.25i)21-s − 1.78i·23-s + 0.200·25-s + (−0.832 − 0.553i)27-s − 0.290i·29-s − 1.53i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09627563636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09627563636\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.14 + 1.30i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (6.11 + 5.43i)T \) |
| good | 7 | \( 1 - 5.02iT - 7T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 13 | \( 1 - 4.52iT - 13T^{2} \) |
| 17 | \( 1 - 1.07iT - 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 + 8.57iT - 23T^{2} \) |
| 29 | \( 1 + 1.56iT - 29T^{2} \) |
| 31 | \( 1 + 8.54iT - 31T^{2} \) |
| 37 | \( 1 - 0.199T + 37T^{2} \) |
| 41 | \( 1 + 4.59T + 41T^{2} \) |
| 43 | \( 1 - 10.9iT - 43T^{2} \) |
| 47 | \( 1 - 8.77iT - 47T^{2} \) |
| 53 | \( 1 + 2.42T + 53T^{2} \) |
| 59 | \( 1 + 12.5iT - 59T^{2} \) |
| 61 | \( 1 - 0.619iT - 61T^{2} \) |
| 71 | \( 1 - 5.87iT - 71T^{2} \) |
| 73 | \( 1 + 5.16T + 73T^{2} \) |
| 79 | \( 1 + 14.6iT - 79T^{2} \) |
| 83 | \( 1 + 4.51iT - 83T^{2} \) |
| 89 | \( 1 + 16.9iT - 89T^{2} \) |
| 97 | \( 1 - 13.3iT - 97T^{2} \) |
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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
−8.026935695023650607238047932133, −7.68441129207008395677237154160, −6.43904131504670805399767435367, −6.16116758303005192641994610102, −5.10627008941618159162973158557, −4.31279155168611879087994070607, −2.99862410650808924492771023957, −2.54472834138889570081069520296, −1.76764077529597207845407569653, −0.02432756858063271635732223445,
1.30647344851870573546081535833, 2.84560485856899923110119342716, 3.46300892466837044757915400384, 4.02802998344659468905390778943, 5.18005465152389874426498930957, 5.32686745095872848476724925800, 7.06198137852938796221247560107, 7.41920166165675294164401884876, 7.988657485646950604149518008590, 8.637911248503813613522363666934
