L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (1.01 − 1.99i)5-s + (2.58 − 0.576i)7-s + (0.707 − 0.707i)8-s + (0.469 − 2.18i)10-s + (−1.41 + 0.819i)11-s + (1 + i)13-s + (2.34 − 1.22i)14-s + (0.500 − 0.866i)16-s + (−0.599 + 2.23i)17-s + (−0.274 − 0.158i)19-s + (−0.111 − 2.23i)20-s + (−1.15 + 1.15i)22-s + (−2.15 − 8.03i)23-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.456 − 0.889i)5-s + (0.976 − 0.217i)7-s + (0.249 − 0.249i)8-s + (0.148 − 0.691i)10-s + (−0.427 + 0.246i)11-s + (0.277 + 0.277i)13-s + (0.626 − 0.327i)14-s + (0.125 − 0.216i)16-s + (−0.145 + 0.542i)17-s + (−0.0629 − 0.0363i)19-s + (−0.0250 − 0.499i)20-s + (−0.246 + 0.246i)22-s + (−0.448 − 1.67i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.29472 - 1.12857i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.29472 - 1.12857i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.01 + 1.99i)T \) |
| 7 | \( 1 + (-2.58 + 0.576i)T \) |
good | 11 | \( 1 + (1.41 - 0.819i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.599 - 2.23i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.274 + 0.158i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.15 + 8.03i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 1.19T + 29T^{2} \) |
| 31 | \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.83 - 6.83i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 2.82iT - 41T^{2} \) |
| 43 | \( 1 + (-5.63 - 5.63i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.10 + 1.63i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (11.2 + 3.01i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-1.04 - 1.80i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.158 + 0.274i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.59 - 0.963i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 7.51iT - 71T^{2} \) |
| 73 | \( 1 + (3.41 - 12.7i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (8.34 + 4.81i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (10.0 - 10.0i)T - 83iT^{2} \) |
| 89 | \( 1 + (7.74 - 13.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.15 - 9.15i)T - 97iT^{2} \) |
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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
−10.59550529699837450129710844721, −9.753040258608655437207571904003, −8.571109147851395190564645070929, −8.001869300267005768218487758317, −6.68399858447868677908770933347, −5.74107618688116003675633281045, −4.74780499963722745371987161345, −4.21768266957564393589787700439, −2.48715142615147256742866955936, −1.33705099849698325564488505438,
1.91367978868100126529304274770, 3.00682440322579936456714423795, 4.17700624673890474811677092375, 5.47433936394455807086155332546, 5.90835651911394257784425480087, 7.25998519926867261483044130738, 7.75147740149831658196861510094, 8.956435892488063674325181752860, 10.01084103041552741389697562862, 11.03524395987225276406072458596