| L(s) = 1 | + (2.23 − 0.123i)5-s − 2.78i·7-s − 3.56·11-s − 0.895i·13-s − 6.09i·17-s − 4.09·19-s − 1.24i·23-s + (4.96 − 0.551i)25-s − 5.45·29-s − 5.84·31-s + (−0.344 − 6.21i)35-s + 5.56i·37-s + 1.68·41-s + 3.47i·43-s + 11.6i·47-s + ⋯ |
| L(s) = 1 | + (0.998 − 0.0552i)5-s − 1.05i·7-s − 1.07·11-s − 0.248i·13-s − 1.47i·17-s − 0.939·19-s − 0.260i·23-s + (0.993 − 0.110i)25-s − 1.01·29-s − 1.04·31-s + (−0.0581 − 1.05i)35-s + 0.915i·37-s + 0.262·41-s + 0.529i·43-s + 1.70i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6881755947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6881755947\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 + 0.123i)T \) |
| good | 7 | \( 1 + 2.78iT - 7T^{2} \) |
| 11 | \( 1 + 3.56T + 11T^{2} \) |
| 13 | \( 1 + 0.895iT - 13T^{2} \) |
| 17 | \( 1 + 6.09iT - 17T^{2} \) |
| 19 | \( 1 + 4.09T + 19T^{2} \) |
| 23 | \( 1 + 1.24iT - 23T^{2} \) |
| 29 | \( 1 + 5.45T + 29T^{2} \) |
| 31 | \( 1 + 5.84T + 31T^{2} \) |
| 37 | \( 1 - 5.56iT - 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 - 3.47iT - 43T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 + 3.84iT - 53T^{2} \) |
| 59 | \( 1 + 9.82T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 7.13iT - 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 7.34iT - 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 7.68iT - 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 - 15.6iT - 97T^{2} \) |
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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
−7.77112623652046317253480960386, −7.43914855980853003711034356819, −6.52828161796189975167492096133, −5.82925564401819673152624396766, −4.98422605303146698086826873261, −4.44130231926528224690031567059, −3.21368637486474667807423478967, −2.49318736931207973407971454763, −1.42180053110679448465771719084, −0.17135982159224845243784859234,
1.82593503286141099287454460845, 2.17388668006426751805487223751, 3.23101071346928614248929064526, 4.25895241397653762159091165119, 5.34572552125355826605778960530, 5.72391979759181743006307623421, 6.32432273378614149493534423300, 7.29445398289900980059787820370, 8.092533952545552674835001901940, 8.939545464484431481990907104833
