L(s) = 1 | + (−0.648 − 0.173i)2-s + (−1.34 − 0.774i)4-s + (−2.13 + 0.672i)5-s + (2.57 + 0.588i)7-s + (1.68 + 1.68i)8-s + (1.49 − 0.0654i)10-s + (−0.0701 + 0.121i)11-s + (−2.35 + 2.35i)13-s + (−1.56 − 0.829i)14-s + (0.750 + 1.29i)16-s + (7.37 − 1.97i)17-s + (3.89 + 6.74i)19-s + (3.38 + 0.750i)20-s + (0.0665 − 0.0665i)22-s + (−0.671 + 2.50i)23-s + ⋯ |
L(s) = 1 | + (−0.458 − 0.122i)2-s + (−0.670 − 0.387i)4-s + (−0.953 + 0.300i)5-s + (0.974 + 0.222i)7-s + (0.595 + 0.595i)8-s + (0.474 − 0.0206i)10-s + (−0.0211 + 0.0366i)11-s + (−0.653 + 0.653i)13-s + (−0.419 − 0.221i)14-s + (0.187 + 0.324i)16-s + (1.78 − 0.478i)17-s + (0.893 + 1.54i)19-s + (0.756 + 0.167i)20-s + (0.0141 − 0.0141i)22-s + (−0.140 + 0.522i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.737395 + 0.269734i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.737395 + 0.269734i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.13 - 0.672i)T \) |
| 7 | \( 1 + (-2.57 - 0.588i)T \) |
good | 2 | \( 1 + (0.648 + 0.173i)T + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (0.0701 - 0.121i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.35 - 2.35i)T - 13iT^{2} \) |
| 17 | \( 1 + (-7.37 + 1.97i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.89 - 6.74i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.671 - 2.50i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 5.09iT - 29T^{2} \) |
| 31 | \( 1 + (-2.54 - 1.46i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (5.73 + 1.53i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 0.261iT - 41T^{2} \) |
| 43 | \( 1 + (2.11 + 2.11i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.402 - 1.50i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.79 + 0.749i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (4.37 - 7.57i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.76 - 2.75i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.02 + 7.54i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 3.56T + 71T^{2} \) |
| 73 | \( 1 + (0.847 + 3.16i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (0.113 - 0.0656i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.33 + 7.33i)T - 83iT^{2} \) |
| 89 | \( 1 + (-2.44 - 4.23i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.25 - 1.25i)T + 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.93691570195353851361360791298, −10.69481192439341805770259969881, −9.957056470613962679349354228068, −8.925535079044114357168367466112, −7.909679363789581250098098962627, −7.42828635551911669729584152089, −5.58757132258976225767011797122, −4.74565777416490280856084075536, −3.47336142711360840926588113740, −1.43357649712107830218761120309,
0.798647395128656582288721964777, 3.26301566689157252147173195054, 4.49372929010907666402006408570, 5.26184567493447286141265548842, 7.24519414898199204559213839457, 7.87565557013275775051368148246, 8.469127610701767490619179732219, 9.607232220655707929041803783345, 10.52655998065656275308497424529, 11.68850477576925855309004196402