| L(s) = 1 | + i·2-s − 4-s + 0.765i·5-s + (−2.29 + 1.32i)7-s − i·8-s − 0.765·10-s − 4.09·11-s − 1.23·13-s + (−1.32 − 2.29i)14-s + 16-s − 3.96i·17-s + (−3.29 + 2.84i)19-s − 0.765i·20-s − 4.09i·22-s − 2.23·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.342i·5-s + (−0.866 + 0.499i)7-s − 0.353i·8-s − 0.242·10-s − 1.23·11-s − 0.342·13-s + (−0.353 − 0.612i)14-s + 0.250·16-s − 0.961i·17-s + (−0.757 + 0.653i)19-s − 0.171i·20-s − 0.872i·22-s − 0.466·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7495613724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7495613724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.29 - 1.32i)T \) |
| 19 | \( 1 + (3.29 - 2.84i)T \) |
| good | 5 | \( 1 - 0.765iT - 5T^{2} \) |
| 11 | \( 1 + 4.09T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 3.96iT - 17T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 - 3.54iT - 29T^{2} \) |
| 31 | \( 1 - 8.42T + 31T^{2} \) |
| 37 | \( 1 + 5.83iT - 37T^{2} \) |
| 41 | \( 1 + 5.62T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 10.9iT - 47T^{2} \) |
| 53 | \( 1 + 8.28iT - 53T^{2} \) |
| 59 | \( 1 - 6.22T + 59T^{2} \) |
| 61 | \( 1 + 5.64iT - 61T^{2} \) |
| 67 | \( 1 - 13.7iT - 67T^{2} \) |
| 71 | \( 1 - 11.2iT - 71T^{2} \) |
| 73 | \( 1 + 9.01iT - 73T^{2} \) |
| 79 | \( 1 - 6.21iT - 79T^{2} \) |
| 83 | \( 1 + 2.03iT - 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 1.90T + 97T^{2} \) |
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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
−8.690345037351204564184991890898, −8.231715131583427822591176999596, −7.16140689786956892707537544540, −6.78133211081716166615232298866, −5.75243910727361960413994869243, −5.24510430269697950778429462314, −4.20003865919042430236639107114, −3.07480238590502117185648807097, −2.35272782423698238980004594622, −0.31444725627637257659035117170,
0.919923516567263259324005492257, 2.39795802010697425164641183164, 3.05577708831749563734390528202, 4.22338335293343525564631846624, 4.76366769202355218461140162084, 5.88633755474303710449377088171, 6.59425678838773058898433520957, 7.67886038889719746143757987655, 8.290630542206949701786192490857, 9.131577040596995871538362377372
