L(s) = 1 | + (−1.25 − 0.656i)2-s + (0.683 − 0.683i)3-s + (1.13 + 1.64i)4-s + (−2.03 + 0.917i)5-s + (−1.30 + 0.407i)6-s + (−0.707 + 0.707i)7-s + (−0.348 − 2.80i)8-s + 2.06i·9-s + (3.15 + 0.189i)10-s + 5.41·11-s + (1.90 + 0.345i)12-s + (3.15 + 3.15i)13-s + (1.34 − 0.421i)14-s + (−0.766 + 2.01i)15-s + (−1.40 + 3.74i)16-s + (−2.46 − 2.46i)17-s + ⋯ |
L(s) = 1 | + (−0.885 − 0.463i)2-s + (0.394 − 0.394i)3-s + (0.569 + 0.822i)4-s + (−0.912 + 0.410i)5-s + (−0.532 + 0.166i)6-s + (−0.267 + 0.267i)7-s + (−0.123 − 0.992i)8-s + 0.688i·9-s + (0.998 + 0.0598i)10-s + 1.63·11-s + (0.548 + 0.0996i)12-s + (0.875 + 0.875i)13-s + (0.360 − 0.112i)14-s + (−0.197 + 0.521i)15-s + (−0.351 + 0.936i)16-s + (−0.598 − 0.598i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.851749 + 0.134907i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.851749 + 0.134907i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.25 + 0.656i)T \) |
| 5 | \( 1 + (2.03 - 0.917i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.683 + 0.683i)T - 3iT^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 + (-3.15 - 3.15i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.46 + 2.46i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.87iT - 19T^{2} \) |
| 23 | \( 1 + (-0.659 - 0.659i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.76T + 29T^{2} \) |
| 31 | \( 1 - 3.09iT - 31T^{2} \) |
| 37 | \( 1 + (0.680 - 0.680i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.39T + 41T^{2} \) |
| 43 | \( 1 + (0.0846 - 0.0846i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.32 + 2.32i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.08 + 8.08i)T + 53iT^{2} \) |
| 59 | \( 1 - 7.94iT - 59T^{2} \) |
| 61 | \( 1 + 4.20iT - 61T^{2} \) |
| 67 | \( 1 + (-7.82 - 7.82i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.85iT - 71T^{2} \) |
| 73 | \( 1 + (-1.45 + 1.45i)T - 73iT^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + (3.88 - 3.88i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.96iT - 89T^{2} \) |
| 97 | \( 1 + (13.2 + 13.2i)T + 97iT^{2} \) |
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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
−11.69277069457219923865817648098, −11.13772877869761590954002676178, −9.982450055733837598258722135078, −8.804749851671768721319315484862, −8.359774120972366593136504155519, −7.08693940566252131331502559934, −6.54561945410641917613638859503, −4.21990151306334850647886007083, −3.16171272322047931779704142467, −1.62506682308402537096665182376,
0.949086252213885292677386914826, 3.36049095853522551915681154629, 4.46384644656505040968391172656, 6.19924508119771772542687442467, 6.94704161642278940006152675890, 8.285259262702087524864947720833, 8.835381575189630091531023331070, 9.612153369471544425571301509230, 10.78262671934816982373318705364, 11.57650759368905770117283504652