| L(s) = 1 | + 2-s − 3-s + 4-s − 2.23·5-s − 6-s − 7-s + 8-s − 2·9-s − 2.23·10-s + 4.23·11-s − 12-s − 1.23·13-s − 14-s + 2.23·15-s + 16-s − 0.763·17-s − 2·18-s + 3·19-s − 2.23·20-s + 21-s + 4.23·22-s − 6.23·23-s − 24-s − 1.23·26-s + 5·27-s − 28-s + 3.47·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.999·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s − 0.666·9-s − 0.707·10-s + 1.27·11-s − 0.288·12-s − 0.342·13-s − 0.267·14-s + 0.577·15-s + 0.250·16-s − 0.185·17-s − 0.471·18-s + 0.688·19-s − 0.499·20-s + 0.218·21-s + 0.903·22-s − 1.30·23-s − 0.204·24-s − 0.242·26-s + 0.962·27-s − 0.188·28-s + 0.644·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 - 3.47T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 6.94T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 + 9.70T + 71T^{2} \) |
| 73 | \( 1 + 2.76T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 6.47T + 89T^{2} \) |
| 97 | \( 1 - 4.70T + 97T^{2} \) |
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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
−7.60509530807806939931715273721, −6.87559956257109589561844998795, −6.10383951980875863985090685867, −5.78563298808023909208452403107, −4.56660702233775397931602344751, −4.24152183511787127457770072620, −3.34472752410415697032471886862, −2.62127165147352686902719652771, −1.22632648015447078784130530159, 0,
1.22632648015447078784130530159, 2.62127165147352686902719652771, 3.34472752410415697032471886862, 4.24152183511787127457770072620, 4.56660702233775397931602344751, 5.78563298808023909208452403107, 6.10383951980875863985090685867, 6.87559956257109589561844998795, 7.60509530807806939931715273721
