L(s) = 1 | + (−0.776 − 0.252i)3-s + (−4.16 + 10.3i)5-s − 1.56i·7-s + (−21.3 − 15.4i)9-s + (−28.6 + 20.8i)11-s + (−34.1 + 46.9i)13-s + (5.85 − 7.00i)15-s + (−21.7 + 7.06i)17-s + (44.6 + 137. i)19-s + (−0.395 + 1.21i)21-s + (−78.6 − 108. i)23-s + (−90.2 − 86.5i)25-s + (25.6 + 35.2i)27-s + (15.4 − 47.3i)29-s + (46.9 + 144. i)31-s + ⋯ |
L(s) = 1 | + (−0.149 − 0.0485i)3-s + (−0.372 + 0.927i)5-s − 0.0845i·7-s + (−0.789 − 0.573i)9-s + (−0.786 + 0.571i)11-s + (−0.727 + 1.00i)13-s + (0.100 − 0.120i)15-s + (−0.310 + 0.100i)17-s + (0.538 + 1.65i)19-s + (−0.00410 + 0.0126i)21-s + (−0.713 − 0.982i)23-s + (−0.721 − 0.692i)25-s + (0.182 + 0.251i)27-s + (0.0986 − 0.303i)29-s + (0.272 + 0.837i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.190533 + 0.576150i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.190533 + 0.576150i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (4.16 - 10.3i)T \) |
good | 3 | \( 1 + (0.776 + 0.252i)T + (21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 + 1.56iT - 343T^{2} \) |
| 11 | \( 1 + (28.6 - 20.8i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (34.1 - 46.9i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (21.7 - 7.06i)T + (3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-44.6 - 137. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (78.6 + 108. i)T + (-3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-15.4 + 47.3i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-46.9 - 144. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-246. + 338. i)T + (-1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (8.27 + 6.00i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 151. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-179. - 58.2i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (506. + 164. i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-448. - 325. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (434. - 315. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-620. + 201. i)T + (2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (192. - 593. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (180. + 249. i)T + (-1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (201. - 621. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (1.00e3 - 326. i)T + (4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (190. - 138. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (427. + 138. i)T + (7.38e5 + 5.36e5i)T^{2} \) |
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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
−14.21767284006455705396177958334, −12.49376360738447377756706771185, −11.76297300861990941463610627284, −10.63630291920037874486786328022, −9.655411302636294392385569924923, −8.139367211138630718026434823580, −7.04593139865157494923629904794, −5.88113018480938639626774352332, −4.12253728849335555392635428519, −2.52178374870900112819189346208,
0.33178152160407724408039179706, 2.82277256337109911458855418928, 4.81933491493819373274383176453, 5.65755101849925162664006949613, 7.60862844172530748848941584546, 8.428407507058787916450179416284, 9.649989257906208322437248303634, 11.01761299006516377152448835287, 11.83775451764074796733348407744, 13.08152901959412801592613384713