L(s) = 1 | + (0.5 + 0.866i)3-s − 3.73i·5-s + (−2.36 − 1.36i)7-s + (−0.499 + 0.866i)9-s + (−1.09 + 0.633i)11-s + (−2.59 − 2.5i)13-s + (3.23 − 1.86i)15-s + (−2.86 + 4.96i)17-s + (−4.09 − 2.36i)19-s − 2.73i·21-s + (−2.09 − 3.63i)23-s − 8.92·25-s − 0.999·27-s + (2.23 + 3.86i)29-s + 1.46i·31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s − 1.66i·5-s + (−0.894 − 0.516i)7-s + (−0.166 + 0.288i)9-s + (−0.331 + 0.191i)11-s + (−0.720 − 0.693i)13-s + (0.834 − 0.481i)15-s + (−0.695 + 1.20i)17-s + (−0.940 − 0.542i)19-s − 0.596i·21-s + (−0.437 − 0.757i)23-s − 1.78·25-s − 0.192·27-s + (0.414 + 0.717i)29-s + 0.262i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.305857 - 0.731761i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.305857 - 0.731761i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (2.59 + 2.5i)T \) |
good | 5 | \( 1 + 3.73iT - 5T^{2} \) |
| 7 | \( 1 + (2.36 + 1.36i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.09 - 0.633i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.86 - 4.96i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.09 + 2.36i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.09 + 3.63i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.23 - 3.86i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.46iT - 31T^{2} \) |
| 37 | \( 1 + (-3.06 + 1.76i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.13 + 4.69i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.83 + 8.36i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 2.19iT - 47T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 + (-6.92 - 4i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.59 + 7.96i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.3 + 6.56i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.09 + 2.36i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6.26iT - 73T^{2} \) |
| 79 | \( 1 - 2.53T + 79T^{2} \) |
| 83 | \( 1 + 0.196iT - 83T^{2} \) |
| 89 | \( 1 + (8.19 - 4.73i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.19 + 3i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.22500011470123770682811914148, −9.329390106688636734482775811878, −8.653634188402530987474949157961, −7.926404970310890597295311601401, −6.65817814973386917527968695676, −5.50608328111035772300690237177, −4.58437145275274747665279125953, −3.84973686827284341600959328169, −2.27895900749039203450240028767, −0.38706085659547321548333630484,
2.42272070900875439720971492115, 2.83950675583261077305839377604, 4.18557868460119040648143612469, 5.89686578922987762295788452171, 6.55623282563754418818297559008, 7.24021351661811350938165437971, 8.126006971168066879377349107495, 9.493213099772387690084741637890, 9.846492301654496377446547903521, 11.09183410343291506671361223744