| L(s) = 1 | + 1.73i·2-s − 0.999·4-s − 2i·7-s + 1.73i·8-s + 3.46·11-s + i·13-s + 3.46·14-s − 5·16-s − 6.92i·17-s − 2·19-s + 5.99i·22-s − 6.92i·23-s − 1.73·26-s + 1.99i·28-s + 6.92·29-s + ⋯ |
| L(s) = 1 | + 1.22i·2-s − 0.499·4-s − 0.755i·7-s + 0.612i·8-s + 1.04·11-s + 0.277i·13-s + 0.925·14-s − 1.25·16-s − 1.68i·17-s − 0.458·19-s + 1.27i·22-s − 1.44i·23-s − 0.339·26-s + 0.377i·28-s + 1.28·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.878150566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.878150566\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 - 1.73iT - 2T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 17 | \( 1 + 6.92iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.693450383298960333431445534765, −7.897187881370745651561862432396, −7.11610376592878259770260115461, −6.63493031318373810083624071450, −6.11617910835117565393654758465, −4.79307103875145028552725567594, −4.58582551978435087327153162889, −3.28989431015129472861471038317, −2.14719474997115338030361967289, −0.64296612699810335872346788223,
1.20731590201477604696342268204, 1.93990995362286380185102193009, 2.96669207862143173750172345935, 3.74627089746051257141118827603, 4.45435267427318885017037057862, 5.67804491588762768491590946936, 6.33667521123317372985277593092, 7.12040362577569498340504072822, 8.296144116449264465423739866845, 8.809833773510477672657746695442
