| L(s) = 1 | + (−1.29 + 0.568i)2-s + (−1.02 − 1.33i)3-s + (1.35 − 1.47i)4-s + (−1.35 − 1.04i)5-s + (2.08 + 1.14i)6-s + (2.49 + 0.869i)7-s + (−0.915 + 2.67i)8-s + (0.0411 − 0.153i)9-s + (2.34 + 0.576i)10-s + (−2.95 − 0.389i)11-s + (−3.35 − 0.299i)12-s + (−4.73 − 1.96i)13-s + (−3.73 + 0.294i)14-s + 2.88i·15-s + (−0.336 − 3.98i)16-s + (−3.38 − 1.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.915 + 0.402i)2-s + (−0.592 − 0.772i)3-s + (0.676 − 0.736i)4-s + (−0.606 − 0.465i)5-s + (0.852 + 0.468i)6-s + (0.944 + 0.328i)7-s + (−0.323 + 0.946i)8-s + (0.0137 − 0.0512i)9-s + (0.742 + 0.182i)10-s + (−0.892 − 0.117i)11-s + (−0.969 − 0.0863i)12-s + (−1.31 − 0.544i)13-s + (−0.996 + 0.0787i)14-s + 0.744i·15-s + (−0.0840 − 0.996i)16-s + (−0.820 − 0.473i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.121051 - 0.337526i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.121051 - 0.337526i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.29 - 0.568i)T \) |
| 7 | \( 1 + (-2.49 - 0.869i)T \) |
| good | 3 | \( 1 + (1.02 + 1.33i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (1.35 + 1.04i)T + (1.29 + 4.82i)T^{2} \) |
| 11 | \( 1 + (2.95 + 0.389i)T + (10.6 + 2.84i)T^{2} \) |
| 13 | \( 1 + (4.73 + 1.96i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + (3.38 + 1.95i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0265 - 0.201i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (0.677 - 2.52i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.550 + 1.32i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (-1.63 + 2.82i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.57 + 1.97i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (7.75 + 7.75i)T + 41iT^{2} \) |
| 43 | \( 1 + (-4.20 - 10.1i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-4.09 + 2.36i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.39 - 1.23i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (-0.412 + 3.13i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-10.8 + 1.42i)T + (58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (7.78 + 10.1i)T + (-17.3 + 64.7i)T^{2} \) |
| 71 | \( 1 + (-0.0346 + 0.0346i)T - 71iT^{2} \) |
| 73 | \( 1 + (4.47 - 1.20i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.28 + 4.20i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.25 - 2.17i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-2.51 - 0.674i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 4.84T + 97T^{2} \) |
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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
−11.83986941364612969115625489894, −11.02027590013120021637216888635, −9.877642937742139084496166767103, −8.661495586116116389124579829532, −7.77609221667224778277183304262, −7.16170183079871923107321606786, −5.77070491540474460128186873064, −4.85485818792247669453272967907, −2.23375637561228844985959979617, −0.40449214616626493138559718048,
2.30511212610137008392760278431, 4.07784100841650141504656779917, 5.10142813151360879621278599295, 6.95689228642791884561696695564, 7.72219972957811250714856449005, 8.752786423905736308476044011314, 10.16966159349566914528763492016, 10.54336389909523498751770008298, 11.42170471684297784523190254854, 12.06057052201857329961382948511
