L(s) = 1 | + (1.36 − 0.356i)2-s + i·3-s + (1.74 − 0.976i)4-s + (−1.65 + 1.50i)5-s + (0.356 + 1.36i)6-s + (2.07 − 1.64i)7-s + (2.03 − 1.95i)8-s − 9-s + (−1.72 + 2.65i)10-s + 4.97i·11-s + (0.976 + 1.74i)12-s + 5.35·13-s + (2.25 − 2.98i)14-s + (−1.50 − 1.65i)15-s + (2.09 − 3.40i)16-s + 3.34·17-s + ⋯ |
L(s) = 1 | + (0.967 − 0.252i)2-s + 0.577i·3-s + (0.872 − 0.488i)4-s + (−0.738 + 0.673i)5-s + (0.145 + 0.558i)6-s + (0.783 − 0.621i)7-s + (0.721 − 0.692i)8-s − 0.333·9-s + (−0.544 + 0.838i)10-s + 1.49i·11-s + (0.282 + 0.503i)12-s + 1.48·13-s + (0.601 − 0.798i)14-s + (−0.388 − 0.426i)15-s + (0.522 − 0.852i)16-s + 0.810·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.39386 + 0.422307i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.39386 + 0.422307i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 + 0.356i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (1.65 - 1.50i)T \) |
| 7 | \( 1 + (-2.07 + 1.64i)T \) |
good | 11 | \( 1 - 4.97iT - 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 + 2.12T + 19T^{2} \) |
| 23 | \( 1 + 6.71T + 23T^{2} \) |
| 29 | \( 1 + 1.42T + 29T^{2} \) |
| 31 | \( 1 + 5.76T + 31T^{2} \) |
| 37 | \( 1 - 0.864iT - 37T^{2} \) |
| 41 | \( 1 - 1.68iT - 41T^{2} \) |
| 43 | \( 1 + 5.29T + 43T^{2} \) |
| 47 | \( 1 + 8.84iT - 47T^{2} \) |
| 53 | \( 1 + 10.1iT - 53T^{2} \) |
| 59 | \( 1 - 4.51T + 59T^{2} \) |
| 61 | \( 1 + 3.54iT - 61T^{2} \) |
| 67 | \( 1 + 0.310T + 67T^{2} \) |
| 71 | \( 1 + 4.74iT - 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 9.38iT - 79T^{2} \) |
| 83 | \( 1 - 6.05iT - 83T^{2} \) |
| 89 | \( 1 + 3.30iT - 89T^{2} \) |
| 97 | \( 1 + 5.32T + 97T^{2} \) |
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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
−11.34544022296546210036035376602, −10.49574086172680161743674775295, −9.966272689200275005596056494137, −8.276133156410522377178273454623, −7.41032890776208585527674338468, −6.46369513345088988840256317679, −5.20181028304608369764060100563, −4.10493888724063172695236003405, −3.63203001898718256584101496291, −1.90208516415405285855456577899,
1.50829304025938536627829137672, 3.24380523043590792118196974501, 4.21946431398531367371999530793, 5.64669499529936576170098040503, 5.99320744288814134986456920920, 7.52946236487177006744600353891, 8.297922666060738865128105234282, 8.748129844981852513134347088705, 10.83799175581887366882061026208, 11.41739607179834537805877008290