| L(s) = 1 | − 2.43·2-s + 3.93·4-s + 3.79i·5-s + 0.367i·7-s − 4.71·8-s − 9.24i·10-s − 0.948i·13-s − 0.896i·14-s + 3.61·16-s − 3.43·17-s + 4.26i·19-s + 14.9i·20-s + 4.96i·23-s − 9.40·25-s + 2.31i·26-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.96·4-s + 1.69i·5-s + 0.139i·7-s − 1.66·8-s − 2.92i·10-s − 0.263i·13-s − 0.239i·14-s + 0.904·16-s − 0.833·17-s + 0.977i·19-s + 3.34i·20-s + 1.03i·23-s − 1.88·25-s + 0.453i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2902085954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2902085954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 5 | \( 1 - 3.79iT - 5T^{2} \) |
| 7 | \( 1 - 0.367iT - 7T^{2} \) |
| 13 | \( 1 + 0.948iT - 13T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 19 | \( 1 - 4.26iT - 19T^{2} \) |
| 23 | \( 1 - 4.96iT - 23T^{2} \) |
| 29 | \( 1 + 2.48T + 29T^{2} \) |
| 31 | \( 1 - 3.51T + 31T^{2} \) |
| 37 | \( 1 + 7.18T + 37T^{2} \) |
| 41 | \( 1 + 2.71T + 41T^{2} \) |
| 43 | \( 1 + 1.88iT - 43T^{2} \) |
| 47 | \( 1 - 0.0206iT - 47T^{2} \) |
| 53 | \( 1 + 5.54iT - 53T^{2} \) |
| 59 | \( 1 + 6.62iT - 59T^{2} \) |
| 61 | \( 1 - 9.75iT - 61T^{2} \) |
| 67 | \( 1 - 4.46T + 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 - 4.39iT - 73T^{2} \) |
| 79 | \( 1 - 10.9iT - 79T^{2} \) |
| 83 | \( 1 + 9.18T + 83T^{2} \) |
| 89 | \( 1 + 3.04iT - 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−10.19116671087564725155100742111, −9.659825728375226606288910237872, −8.660251266215841459501006329769, −7.88087598729276182783857001913, −7.12191493927450181128236273420, −6.60631919215030036856128197388, −5.63560395915246864087765718958, −3.74460925830925112486475350559, −2.71084964315007959743580427594, −1.76134495542290762331392165882,
0.23600968729111307105706260439, 1.30492980399026857215308234693, 2.41893417288374219102995254190, 4.24956708819211652331820602001, 5.10226307252524744673554936567, 6.36938163211037379726028965745, 7.20852008654909486287121125540, 8.157391774747920734857841004557, 8.816613948352231354422520921037, 9.098465762272512833095743770795
