| L(s) = 1 | − 2i·7-s + 2·11-s − 4i·13-s + 2i·17-s + 4·19-s − 8i·23-s − 10·29-s − 4·31-s − 8i·43-s + 8i·47-s + 3·49-s + 6i·53-s + 14·59-s − 14·61-s + 4i·67-s + ⋯ |
| L(s) = 1 | − 0.755i·7-s + 0.603·11-s − 1.10i·13-s + 0.485i·17-s + 0.917·19-s − 1.66i·23-s − 1.85·29-s − 0.718·31-s − 1.21i·43-s + 1.16i·47-s + 0.428·49-s + 0.824i·53-s + 1.82·59-s − 1.79·61-s + 0.488i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.403338159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403338159\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.283029875137552539839404957429, −7.47940585717110053184105306561, −7.00785676840452574169044191781, −5.98492367549555026670610042290, −5.40653079398212146529516682529, −4.32516884805919619448121987457, −3.70545176466988432949125348599, −2.77900230776342941726089322719, −1.53868997527537161760754976390, −0.42275476545506171524972591216,
1.39474377658818006533583857835, 2.23174150751661651254967236708, 3.41189843904115168830870708221, 4.04105233654486842079248085501, 5.24981477649923373428271719079, 5.61321710272776980579298178670, 6.66116886536566051070794851904, 7.26920875365545365183469001472, 8.002696829222782484910162859193, 9.135621494987194465069505271885
