L(s) = 1 | + (0.474 − 0.127i)2-s + (−0.708 − 1.58i)3-s + (−1.52 + 0.879i)4-s + (2.23 − 0.0580i)5-s + (−0.536 − 0.659i)6-s + (−1.30 + 1.30i)8-s + (−1.99 + 2.24i)9-s + (1.05 − 0.311i)10-s + (2.31 − 1.33i)11-s + (2.46 + 1.78i)12-s + (2.14 + 2.14i)13-s + (−1.67 − 3.49i)15-s + (1.30 − 2.26i)16-s + (1.19 − 4.46i)17-s + (−0.661 + 1.31i)18-s + (4.54 + 2.62i)19-s + ⋯ |
L(s) = 1 | + (0.335 − 0.0898i)2-s + (−0.409 − 0.912i)3-s + (−0.761 + 0.439i)4-s + (0.999 − 0.0259i)5-s + (−0.219 − 0.269i)6-s + (−0.461 + 0.461i)8-s + (−0.665 + 0.746i)9-s + (0.332 − 0.0984i)10-s + (0.697 − 0.402i)11-s + (0.712 + 0.515i)12-s + (0.596 + 0.596i)13-s + (−0.432 − 0.901i)15-s + (0.326 − 0.565i)16-s + (0.290 − 1.08i)17-s + (−0.155 + 0.310i)18-s + (1.04 + 0.601i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49802 - 0.595201i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49802 - 0.595201i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.708 + 1.58i)T \) |
| 5 | \( 1 + (-2.23 + 0.0580i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.474 + 0.127i)T + (1.73 - i)T^{2} \) |
| 11 | \( 1 + (-2.31 + 1.33i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.14 - 2.14i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.19 + 4.46i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.54 - 2.62i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.932 + 3.48i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 + (-2.64 - 4.57i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.784 + 2.92i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 11.5iT - 41T^{2} \) |
| 43 | \( 1 + (-0.759 - 0.759i)T + 43iT^{2} \) |
| 47 | \( 1 + (-10.4 + 2.80i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (6.05 + 1.62i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (0.0797 + 0.138i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.36 - 4.09i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.39 - 1.98i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (-1.52 + 5.68i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.37 - 1.94i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.03 + 4.03i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.97 - 3.42i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.86 - 1.86i)T - 97iT^{2} \) |
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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
−10.29460804612015160749886192848, −9.188948947205936149449058316280, −8.735788714430572272747987232035, −7.58375494082894166080952584933, −6.66341359990452169550948604802, −5.72792814759312321033816875952, −5.09549307710838837779460233306, −3.72326578309236138088048110574, −2.47982367636439365099120992753, −1.04237220852906803318795848070,
1.22782749029391226382757816674, 3.21790093180267437746353058674, 4.19476453200667753093529665926, 5.15403289054636408108985123042, 5.86197957089076250854312811733, 6.46602662494152808808894082483, 8.103590264162310022408699573751, 9.292463844859062316020630414572, 9.512157209194808185361484341032, 10.33403708902887463163437003073