| L(s) = 1 | + 4.75i·2-s − 3i·3-s − 14.5·4-s + 14.2·6-s + 7i·7-s − 31.3i·8-s − 9·9-s + 7.31·11-s + 43.7i·12-s + 4.15i·13-s − 33.2·14-s + 32.2·16-s + 53.5i·17-s − 42.7i·18-s − 88.9·19-s + ⋯ |
| L(s) = 1 | + 1.68i·2-s − 0.577i·3-s − 1.82·4-s + 0.970·6-s + 0.377i·7-s − 1.38i·8-s − 0.333·9-s + 0.200·11-s + 1.05i·12-s + 0.0886i·13-s − 0.635·14-s + 0.504·16-s + 0.763i·17-s − 0.560i·18-s − 1.07·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6206256510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6206256510\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
| good | 2 | \( 1 - 4.75iT - 8T^{2} \) |
| 11 | \( 1 - 7.31T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.15iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 53.5iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 88.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 156. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 42.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 14.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 293. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 127.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 210. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 468. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 115. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 314.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 768.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 717. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 737.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 477. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 279.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 776. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 29.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 231. iT - 9.12e5T^{2} \) |
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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
−10.17255154795484161510023740489, −8.863496149718666683527584333802, −8.500330461065653271539254099275, −7.55817704853450836948200971381, −6.61902686314583351492931163019, −6.11089065184140298753238815991, −5.08889516319229655421421417222, −3.97779587724282728837900069065, −2.14784629872188727054898733758, −0.20225152838630490183475855822,
1.23463191518696714081424709739, 2.55987550415984747265634936173, 3.59979443108973621092505945252, 4.38259451582036348370422099189, 5.41542306951324260293949494732, 6.90104525502397096201697888253, 8.251971580038313779551792071454, 9.207726907950561514173280584804, 9.857249432177349047202087973436, 10.56734144484390345361581683593
