| L(s) = 1 | − 0.732i·5-s − 7-s − 1.46i·11-s + 3.26i·13-s − 2·17-s + 4.73i·19-s + 3.46·23-s + 4.46·25-s − 5.46i·29-s − 4·31-s + 0.732i·35-s − 5.46i·37-s + 2·41-s + 1.46i·43-s − 10.9·47-s + ⋯ |
| L(s) = 1 | − 0.327i·5-s − 0.377·7-s − 0.441i·11-s + 0.906i·13-s − 0.485·17-s + 1.08i·19-s + 0.722·23-s + 0.892·25-s − 1.01i·29-s − 0.718·31-s + 0.123i·35-s − 0.898i·37-s + 0.312·41-s + 0.223i·43-s − 1.59·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.279569950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.279569950\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 0.732iT - 5T^{2} \) |
| 11 | \( 1 + 1.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.26iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4.73iT - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 5.46iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 5.46iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 1.46iT - 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 7.66iT - 59T^{2} \) |
| 61 | \( 1 - 13.1iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 + 2.92T + 79T^{2} \) |
| 83 | \( 1 - 11.6iT - 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−8.844960581044570451190111332249, −7.82586295271407634607811794561, −7.16227701957063019762158053507, −6.30439962037175868079926733380, −5.76350657422559029789120680018, −4.75226825423893273235075553954, −4.06816711069393023174207906793, −3.18347499421894553522735376704, −2.16341679187750412422639067135, −1.06119790628535093346913823740,
0.40776064959521645545769169560, 1.79502251677854224753786018614, 2.94947552660839131081949625188, 3.39314592152252274967408894875, 4.72753801948510179414020189152, 5.12068952021409522697721662617, 6.22892932815375096004856077072, 6.89085266550901358105328698790, 7.37205913130793129631039033610, 8.401660907060080285884545494943
