| L(s) = 1 | − 3i·3-s + 1.64·5-s + (−15.2 + 10.4i)7-s − 9·9-s + 20.3·11-s + 13.0·13-s − 4.92i·15-s + 23.9i·17-s − 87.7i·19-s + (31.4 + 45.8i)21-s + 73.6i·23-s − 122.·25-s + 27i·27-s − 58.9i·29-s + 124.·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.146·5-s + (−0.824 + 0.565i)7-s − 0.333·9-s + 0.557·11-s + 0.279·13-s − 0.0847i·15-s + 0.341i·17-s − 1.05i·19-s + (0.326 + 0.476i)21-s + 0.667i·23-s − 0.978·25-s + 0.192i·27-s − 0.377i·29-s + 0.723·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.025166091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025166091\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 + (15.2 - 10.4i)T \) |
| good | 5 | \( 1 - 1.64T + 125T^{2} \) |
| 11 | \( 1 - 20.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 87.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 73.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 58.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 56.5iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 135. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 259.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 217.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 529. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 685. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 149.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 409.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 885. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 269. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 902. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 623. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 986. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 179. iT - 9.12e5T^{2} \) |
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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
−9.072465341113567565334860136615, −8.124703624292191447551745621527, −7.26895241648783708684616538391, −6.31251497306415632266335184019, −5.96290106333999597588134078866, −4.74334539000873610975285682304, −3.56643333238305816229543500733, −2.65187081726142270776624743731, −1.56503722281491082712169910055, −0.25696968284748953370564800824,
1.07494646629641473226882902636, 2.53673143791863739925948912761, 3.68042106942316805562399419403, 4.17861576931128922839475592410, 5.41970062990660542028902014251, 6.23009209431378274292310386045, 6.97721119076040681799634148352, 7.995242646865038077821090107392, 8.854303388233849300012336119303, 9.694809169906433160830399931823
