L(s) = 1 | + (−1.37 − 0.340i)2-s + (0.176 + 1.11i)3-s + (1.76 + 0.933i)4-s + (0.507 + 2.17i)5-s + (0.136 − 1.59i)6-s − 4.59i·7-s + (−2.11 − 1.88i)8-s + (1.63 − 0.532i)9-s + (0.0439 − 3.16i)10-s + (−2.56 − 5.03i)11-s + (−0.728 + 2.13i)12-s + (1.56 − 3.07i)13-s + (−1.56 + 6.30i)14-s + (−2.33 + 0.950i)15-s + (2.25 + 3.30i)16-s + (0.981 − 0.713i)17-s + ⋯ |
L(s) = 1 | + (−0.970 − 0.240i)2-s + (0.102 + 0.644i)3-s + (0.884 + 0.466i)4-s + (0.226 + 0.973i)5-s + (0.0558 − 0.649i)6-s − 1.73i·7-s + (−0.746 − 0.665i)8-s + (0.546 − 0.177i)9-s + (0.0138 − 0.999i)10-s + (−0.773 − 1.51i)11-s + (−0.210 + 0.617i)12-s + (0.434 − 0.852i)13-s + (−0.417 + 1.68i)14-s + (−0.604 + 0.245i)15-s + (0.564 + 0.825i)16-s + (0.238 − 0.173i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.949538 - 0.241283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.949538 - 0.241283i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.37 + 0.340i)T \) |
| 5 | \( 1 + (-0.507 - 2.17i)T \) |
good | 3 | \( 1 + (-0.176 - 1.11i)T + (-2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + 4.59iT - 7T^{2} \) |
| 11 | \( 1 + (2.56 + 5.03i)T + (-6.46 + 8.89i)T^{2} \) |
| 13 | \( 1 + (-1.56 + 3.07i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.981 + 0.713i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-6.70 - 1.06i)T + (18.0 + 5.87i)T^{2} \) |
| 23 | \( 1 + (1.41 + 0.459i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.08 - 6.83i)T + (-27.5 + 8.96i)T^{2} \) |
| 31 | \( 1 + (-1.40 + 1.01i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (4.04 + 2.06i)T + (21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (1.56 - 0.507i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.39 + 1.39i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.49 + 4.72i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-3.66 + 0.580i)T + (50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-12.0 - 6.12i)T + (34.6 + 47.7i)T^{2} \) |
| 61 | \( 1 + (-1.53 + 0.782i)T + (35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-1.81 - 0.287i)T + (63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (2.40 - 3.31i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-8.13 - 2.64i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (9.12 + 6.62i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (3.18 + 0.505i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (4.23 + 1.37i)T + (72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-3.31 - 2.40i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.74115273720790740300604417255, −10.33944604195197372975781782036, −9.800646948590920720458940937915, −8.440524696804810675003886423394, −7.50618915731938775688269805148, −6.87764664292795183250908064699, −5.55053522503547724400573658362, −3.60973588113733925367977130914, −3.21470391276757278112618149789, −0.958430774024420732333974418882,
1.60079941991860182909438125624, 2.38912436933311429200351253833, 4.84875308108342982833677395166, 5.72097293891843048254286808159, 6.83384464273279427603021239973, 7.85189594534713326581063534829, 8.517356011755757429651764853959, 9.594397870358144025821465051659, 9.861125461922150862977206334302, 11.59836145315416785915743764160