L(s) = 1 | + (−0.538 − 2.01i)2-s + (0.965 + 0.258i)3-s + (−2.02 + 1.16i)4-s − 2.08i·6-s + (−1.74 − 1.99i)7-s + (0.495 + 0.495i)8-s + (0.866 + 0.499i)9-s + (−0.971 − 1.68i)11-s + (−2.25 + 0.604i)12-s + (1.84 − 1.84i)13-s + (−3.06 + 4.57i)14-s + (−1.60 + 2.78i)16-s + (1.62 − 6.06i)17-s + (0.538 − 2.01i)18-s + (−1.50 + 2.61i)19-s + ⋯ |
L(s) = 1 | + (−0.381 − 1.42i)2-s + (0.557 + 0.149i)3-s + (−1.01 + 0.584i)4-s − 0.850i·6-s + (−0.658 − 0.752i)7-s + (0.175 + 0.175i)8-s + (0.288 + 0.166i)9-s + (−0.292 − 0.507i)11-s + (−0.651 + 0.174i)12-s + (0.511 − 0.511i)13-s + (−0.819 + 1.22i)14-s + (−0.401 + 0.695i)16-s + (0.394 − 1.47i)17-s + (0.127 − 0.474i)18-s + (−0.345 + 0.599i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0815412 + 0.998053i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0815412 + 0.998053i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.74 + 1.99i)T \) |
good | 2 | \( 1 + (0.538 + 2.01i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (0.971 + 1.68i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.84 + 1.84i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.62 + 6.06i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.50 - 2.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.35 - 1.43i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 5.27iT - 29T^{2} \) |
| 31 | \( 1 + (5.10 - 2.94i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.241 - 0.901i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 10.2iT - 41T^{2} \) |
| 43 | \( 1 + (6.54 + 6.54i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.75 + 1.00i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.301 - 1.12i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.06 - 5.30i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.50 - 3.75i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-14.3 - 3.85i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + (6.05 + 1.62i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-14.2 - 8.21i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.197 - 0.197i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.08 + 10.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.9 + 10.9i)T + 97iT^{2} \) |
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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
−10.27365457330051688975064932782, −9.842354174943066085337618384159, −8.918573231354605889088846786637, −8.014891888311180446397055514802, −6.93041777918288470786954557836, −5.58046649906365888523432075600, −3.93891866772240239856674184870, −3.38386014351859753515027132115, −2.23527164473182848999212278948, −0.60650925809915933973546240715,
2.16741908479106121379690003865, 3.67047498282342503199841000953, 5.08589322746228395723111280925, 6.22663377449425596259823538692, 6.65950480642430357274987874172, 7.88525336693105466886366789502, 8.414627422938998083087608759125, 9.266294764196588921990994867945, 9.956751136139030197524564496916, 11.26111332649464054611831984283