| L(s) = 1 | + 3.29i·3-s − 1.78i·7-s − 7.87·9-s − 5.08·11-s − 1.29i·13-s − 0.213i·17-s + 19-s + 5.89·21-s − 3.72i·23-s − 16.0i·27-s − 0.870·29-s − 16.7i·33-s + 2i·37-s + 4.27·39-s + 8.59·41-s + ⋯ |
| L(s) = 1 | + 1.90i·3-s − 0.675i·7-s − 2.62·9-s − 1.53·11-s − 0.359i·13-s − 0.0517i·17-s + 0.229·19-s + 1.28·21-s − 0.776i·23-s − 3.09i·27-s − 0.161·29-s − 2.91i·33-s + 0.328i·37-s + 0.684·39-s + 1.34·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.142898191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.142898191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.29iT - 3T^{2} \) |
| 7 | \( 1 + 1.78iT - 7T^{2} \) |
| 11 | \( 1 + 5.08T + 11T^{2} \) |
| 13 | \( 1 + 1.29iT - 13T^{2} \) |
| 17 | \( 1 + 0.213iT - 17T^{2} \) |
| 23 | \( 1 + 3.72iT - 23T^{2} \) |
| 29 | \( 1 + 0.870T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 8.59T + 41T^{2} \) |
| 43 | \( 1 + 3.67iT - 43T^{2} \) |
| 47 | \( 1 - 4.65iT - 47T^{2} \) |
| 53 | \( 1 - 11.0iT - 53T^{2} \) |
| 59 | \( 1 - 4.70T + 59T^{2} \) |
| 61 | \( 1 - 3.51T + 61T^{2} \) |
| 67 | \( 1 + 1.12iT - 67T^{2} \) |
| 71 | \( 1 - 8.76T + 71T^{2} \) |
| 73 | \( 1 - 6.80iT - 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 9.74iT - 83T^{2} \) |
| 89 | \( 1 - 6.76T + 89T^{2} \) |
| 97 | \( 1 + 4.16iT - 97T^{2} \) |
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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
−8.645238315829141040651353748430, −8.011016614926993063084647821514, −7.23023390637691402810923139053, −5.93837314938285479795521522476, −5.42774559470322649013999145821, −4.62215930456886154674755318837, −4.12452443222169320759764751278, −3.14327578219914648328075433019, −2.54285430051668785870194570838, −0.43633270999248396691422672889,
0.813819482521378409310606198258, 2.06849824965731914520627372508, 2.47746876956439835558153654184, 3.46891268047027817947593825152, 5.05764035846719257121564050303, 5.62708923985748856482085766401, 6.25877460940886423416976043402, 7.14400245054850604366597799711, 7.62961825898824289645098686927, 8.242831992292813961791070088434
