| L(s) = 1 | − 0.806i·3-s + (1.48 − 1.67i)5-s + 2.15i·7-s + 2.35·9-s + 0.387·11-s − 0.962i·13-s + (−1.35 − 1.19i)15-s − 1.61i·17-s + 6.31·19-s + 1.73·21-s + 6.15i·23-s + (−0.612 − 4.96i)25-s − 4.31i·27-s + 2·29-s − 9.92·31-s + ⋯ |
| L(s) = 1 | − 0.465i·3-s + (0.662 − 0.749i)5-s + 0.815i·7-s + 0.783·9-s + 0.116·11-s − 0.266i·13-s + (−0.348 − 0.308i)15-s − 0.390i·17-s + 1.44·19-s + 0.379·21-s + 1.28i·23-s + (−0.122 − 0.992i)25-s − 0.829i·27-s + 0.371·29-s − 1.78·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.68564 - 0.638365i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68564 - 0.638365i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.48 + 1.67i)T \) |
| good | 3 | \( 1 + 0.806iT - 3T^{2} \) |
| 7 | \( 1 - 2.15iT - 7T^{2} \) |
| 11 | \( 1 - 0.387T + 11T^{2} \) |
| 13 | \( 1 + 0.962iT - 13T^{2} \) |
| 17 | \( 1 + 1.61iT - 17T^{2} \) |
| 19 | \( 1 - 6.31T + 19T^{2} \) |
| 23 | \( 1 - 6.15iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 + 6.57iT - 37T^{2} \) |
| 41 | \( 1 - 4.57T + 41T^{2} \) |
| 43 | \( 1 + 11.5iT - 43T^{2} \) |
| 47 | \( 1 + 4.54iT - 47T^{2} \) |
| 53 | \( 1 - 8.96iT - 53T^{2} \) |
| 59 | \( 1 + 6.31T + 59T^{2} \) |
| 61 | \( 1 + 0.261T + 61T^{2} \) |
| 67 | \( 1 - 9.89iT - 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 13.0iT - 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 + 2.88iT - 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 9.61iT - 97T^{2} \) |
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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
−10.31788675326365634165781230013, −9.398851117000629196948907733472, −8.987138214793447892650552698178, −7.73154980242870551283503887513, −7.04598185369260497164332157868, −5.67091714590042745504455269431, −5.31651097508777487429847826360, −3.86475817639767833167489728447, −2.34784486574844601187947777581, −1.21745518006587487699192510333,
1.49636986216914167016105981809, 3.05475489488875960938806754662, 4.07984441611559768358598233867, 5.09049726672864314994267789092, 6.32562725806722147463132417096, 7.05714208796549723331363853659, 7.892836790027876105286456397840, 9.336408387567496010974596453694, 9.797574267271871682437689160243, 10.66140106879211154130580358551
