| L(s) = 1 | − i·2-s − 4-s + 2.54i·5-s − i·7-s + i·8-s + 2.54·10-s − 3.33i·11-s + (−3.27 + 1.51i)13-s − 14-s + 16-s + 7.09·17-s + 4.54i·19-s − 2.54i·20-s − 3.33·22-s + 1.75·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.13i·5-s − 0.377i·7-s + 0.353i·8-s + 0.804·10-s − 1.00i·11-s + (−0.907 + 0.419i)13-s − 0.267·14-s + 0.250·16-s + 1.71·17-s + 1.04i·19-s − 0.569i·20-s − 0.710·22-s + 0.366·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.417534131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.417534131\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 13 | \( 1 + (3.27 - 1.51i)T \) |
| good | 5 | \( 1 - 2.54iT - 5T^{2} \) |
| 11 | \( 1 + 3.33iT - 11T^{2} \) |
| 17 | \( 1 - 7.09T + 17T^{2} \) |
| 19 | \( 1 - 4.54iT - 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 + 7.57T + 29T^{2} \) |
| 31 | \( 1 - 3.33iT - 31T^{2} \) |
| 37 | \( 1 + 3.75iT - 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 9.09T + 43T^{2} \) |
| 47 | \( 1 - 11.3iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 8.06iT - 59T^{2} \) |
| 61 | \( 1 + 0.785T + 61T^{2} \) |
| 67 | \( 1 - 5.75iT - 67T^{2} \) |
| 71 | \( 1 - 11.1iT - 71T^{2} \) |
| 73 | \( 1 - 9.33iT - 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 - 11.2iT - 83T^{2} \) |
| 89 | \( 1 + 15.0iT - 89T^{2} \) |
| 97 | \( 1 + 7.75iT - 97T^{2} \) |
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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
−9.729335220416505249324305098991, −8.781814089037556796131262013298, −7.68260955679953935826521609466, −7.28309422848531062502652433407, −6.06770248878088331666782537238, −5.41577668338039580885831538896, −4.08143528449735087195759981861, −3.33079501071606263418679405479, −2.58507684435035833590988941521, −1.18063557624227386495500910813,
0.63019109724969909408816700856, 2.10548935790796452728326761600, 3.51253317612974734145963209624, 4.74364796271819359476741477701, 5.14157187608412471362582694543, 5.90750953326133879694370515052, 7.18639270060927706581343487475, 7.61708007622658875900182328421, 8.497419175252056365387965448392, 9.364583494257564269985298765449
