L(s) = 1 | + (0.309 + 1.37i)2-s + 3-s + (−1.80 + 0.853i)4-s + i·5-s + (0.309 + 1.37i)6-s + (2.64 + 0.0785i)7-s + (−1.73 − 2.23i)8-s + 9-s + (−1.37 + 0.309i)10-s + 0.987i·11-s + (−1.80 + 0.853i)12-s + 4.69i·13-s + (0.709 + 3.67i)14-s + i·15-s + (2.54 − 3.08i)16-s + 3.93i·17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.975i)2-s + 0.577·3-s + (−0.904 + 0.426i)4-s + 0.447i·5-s + (0.126 + 0.563i)6-s + (0.999 + 0.0296i)7-s + (−0.614 − 0.789i)8-s + 0.333·9-s + (−0.436 + 0.0977i)10-s + 0.297i·11-s + (−0.522 + 0.246i)12-s + 1.30i·13-s + (0.189 + 0.981i)14-s + 0.258i·15-s + (0.635 − 0.771i)16-s + 0.954i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.979356 + 1.49540i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.979356 + 1.49540i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 1.37i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-2.64 - 0.0785i)T \) |
good | 11 | \( 1 - 0.987iT - 11T^{2} \) |
| 13 | \( 1 - 4.69iT - 13T^{2} \) |
| 17 | \( 1 - 3.93iT - 17T^{2} \) |
| 19 | \( 1 + 0.223T + 19T^{2} \) |
| 23 | \( 1 + 5.88iT - 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 - 8.26T + 37T^{2} \) |
| 41 | \( 1 + 7.34iT - 41T^{2} \) |
| 43 | \( 1 + 4.32iT - 43T^{2} \) |
| 47 | \( 1 - 2.40T + 47T^{2} \) |
| 53 | \( 1 + 8.35T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 4.93iT - 61T^{2} \) |
| 67 | \( 1 + 7.84iT - 67T^{2} \) |
| 71 | \( 1 - 8.49iT - 71T^{2} \) |
| 73 | \( 1 + 14.4iT - 73T^{2} \) |
| 79 | \( 1 + 11.5iT - 79T^{2} \) |
| 83 | \( 1 + 1.67T + 83T^{2} \) |
| 89 | \( 1 - 0.493iT - 89T^{2} \) |
| 97 | \( 1 + 4.31iT - 97T^{2} \) |
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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
−11.54349795722472866623569215925, −10.47468135026714814033845294069, −9.331296515634044965056690969754, −8.594316186029348601624697621756, −7.72721804731522825725432860855, −6.93853124879651555751218559838, −5.91537550301532491687212267206, −4.59853035664558202451404338840, −3.85227022824274035175973062792, −2.09124470444325012590511219387,
1.16360975656584479367777353260, 2.58519639144915520094548214525, 3.75651303811333661331841845726, 4.91905490190297155200917975973, 5.68765864821129632167421378818, 7.64479543366023548422023508187, 8.244026110896239550879765631940, 9.289415189629131913287578401628, 9.952479639406503996082166850868, 11.18637053154307323589054127686