L(s) = 1 | + 3.87·5-s + (2 − 1.73i)7-s − 2.23i·11-s − 3.46i·13-s + 5.19i·19-s − 2.23i·23-s + 10.0·25-s − 4.47i·29-s − 1.73i·31-s + (7.74 − 6.70i)35-s − 37-s + 3.87·41-s − 2·43-s − 7.74·47-s + (1.00 − 6.92i)49-s + ⋯ |
L(s) = 1 | + 1.73·5-s + (0.755 − 0.654i)7-s − 0.674i·11-s − 0.960i·13-s + 1.19i·19-s − 0.466i·23-s + 2.00·25-s − 0.830i·29-s − 0.311i·31-s + (1.30 − 1.13i)35-s − 0.164·37-s + 0.604·41-s − 0.304·43-s − 1.12·47-s + (0.142 − 0.989i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.880731239\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.880731239\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 - 3.87T + 5T^{2} \) |
| 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 + 2.23iT - 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 3.87T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 7.74T + 47T^{2} \) |
| 53 | \( 1 - 8.94iT - 53T^{2} \) |
| 59 | \( 1 - 7.74T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 - 11.1iT - 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 + 7.74T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.459602225507776480902592475863, −8.076978817456478406480305257709, −7.05183964054449592016492604359, −6.12620587509765358537298960920, −5.67649203532205639063722907319, −4.94093571781666379173901085359, −3.87974922662895039700121734281, −2.79017399054597350358773250514, −1.85931143963676880871271953232, −0.930789214913597015010690889754,
1.47473377615761563774349049437, 2.06152133928219905593189017382, 2.88614031195433916707824735795, 4.37765547575386927378736418760, 5.15271720385637278652934674306, 5.60391034936522234635863257996, 6.66948538371421514004885982389, 7.01884987463605748443026353290, 8.294242166855991558384206552025, 8.970294632130577726678951333616