| L(s) = 1 | − i·2-s − 4-s + (0.301 − 2.21i)5-s + 3.63i·7-s + i·8-s + (−2.21 − 0.301i)10-s − 2·11-s − i·13-s + 3.63·14-s + 16-s + 6.67i·17-s − 8.06·19-s + (−0.301 + 2.21i)20-s + 2i·22-s + 0.794i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.134 − 0.990i)5-s + 1.37i·7-s + 0.353i·8-s + (−0.700 − 0.0952i)10-s − 0.603·11-s − 0.277i·13-s + 0.971·14-s + 0.250·16-s + 1.61i·17-s − 1.85·19-s + (−0.0673 + 0.495i)20-s + 0.426i·22-s + 0.165i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5818474061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5818474061\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-0.301 + 2.21i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 3.63iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 6.67iT - 17T^{2} \) |
| 19 | \( 1 + 8.06T + 19T^{2} \) |
| 23 | \( 1 - 0.794iT - 23T^{2} \) |
| 29 | \( 1 + 8.06T + 29T^{2} \) |
| 31 | \( 1 - 5.20T + 31T^{2} \) |
| 37 | \( 1 - 0.431iT - 37T^{2} \) |
| 41 | \( 1 + 6.86T + 41T^{2} \) |
| 43 | \( 1 - 5.80iT - 43T^{2} \) |
| 47 | \( 1 - 7.63iT - 47T^{2} \) |
| 53 | \( 1 - 0.794iT - 53T^{2} \) |
| 59 | \( 1 - 8.06T + 59T^{2} \) |
| 61 | \( 1 + 2.86T + 61T^{2} \) |
| 67 | \( 1 + 5.20iT - 67T^{2} \) |
| 71 | \( 1 - 9.46T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 1.93T + 79T^{2} \) |
| 83 | \( 1 - 2.06iT - 83T^{2} \) |
| 89 | \( 1 + 8.06T + 89T^{2} \) |
| 97 | \( 1 + 13.6iT - 97T^{2} \) |
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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
−9.957187421474893396569394703018, −9.108947539511798631468249746726, −8.430303902582524347071372468937, −8.037386774035711170379450815184, −6.27387417945188038345891709133, −5.65574609942550134242670595321, −4.77999458888552860098192226859, −3.82747306314740883253021311495, −2.47188698417116416374993877768, −1.67350836674783665573746645895,
0.23797890330297328424662144559, 2.25221639044164979339684902519, 3.54281237195181493593853758740, 4.39337530530037830456033414902, 5.41054294488955567092301660819, 6.58481354318219511267235277257, 7.01903313544583978133330657228, 7.68342698810026752767188289314, 8.623408397857896240325610250051, 9.715370520374212790699914567366
