| L(s) = 1 | + i·2-s − 1.56i·3-s − 4-s + 1.56·6-s − 3.12i·7-s − i·8-s + 0.561·9-s − 4·11-s + 1.56i·12-s − 0.438i·13-s + 3.12·14-s + 16-s + i·17-s + 0.561i·18-s − 1.56·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.901i·3-s − 0.5·4-s + 0.637·6-s − 1.18i·7-s − 0.353i·8-s + 0.187·9-s − 1.20·11-s + 0.450i·12-s − 0.121i·13-s + 0.834·14-s + 0.250·16-s + 0.242i·17-s + 0.132i·18-s − 0.358·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.472473 - 0.764477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.472473 - 0.764477i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.56iT - 3T^{2} \) |
| 7 | \( 1 + 3.12iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 0.438iT - 13T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 23 | \( 1 + 3.12iT - 23T^{2} \) |
| 29 | \( 1 + 6.68T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 1.12iT - 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 7.12iT - 43T^{2} \) |
| 47 | \( 1 - 2.43iT - 47T^{2} \) |
| 53 | \( 1 + 3.56iT - 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 + 0.876iT - 67T^{2} \) |
| 71 | \( 1 + 16.6T + 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 10.2iT - 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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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
−10.13707671640464827841804855676, −8.740251846585727882982387455954, −7.929341942481105509557404724322, −7.29140338375142982622832240796, −6.74929042748835889138657864642, −5.66923789787085704032810633207, −4.64533597781767906258719247013, −3.58575278116087150754053511930, −1.96410154343783385734264831723, −0.42247611161915083596435617575,
1.95809256434979639500389585690, 3.01503767476057869584094690851, 4.04045539871305934225924780676, 5.15649114311591850240191736452, 5.59465057718043016443589477415, 7.12237449955511959619547328927, 8.252642972966322267530851180805, 9.008954312670953977626615633101, 9.767022107862450685381129200024, 10.34644517047357967337263294183
