| L(s) = 1 | + (−1.03 + 1.88i)2-s + (0.613 − 0.721i)3-s + (−1.40 − 2.22i)4-s + (−0.771 − 0.555i)5-s + (0.723 + 1.90i)6-s + (−2.47 + 0.927i)7-s + (1.36 − 0.0827i)8-s + (0.337 + 2.07i)9-s + (1.84 − 0.877i)10-s + (−0.227 + 0.533i)11-s + (−2.47 − 0.350i)12-s + (−0.397 − 0.756i)13-s + (0.825 − 5.63i)14-s + (−0.874 + 0.215i)15-s + (0.995 − 2.09i)16-s + (0.0659 + 0.322i)17-s + ⋯ |
| L(s) = 1 | + (−0.734 + 1.33i)2-s + (0.354 − 0.416i)3-s + (−0.704 − 1.11i)4-s + (−0.344 − 0.248i)5-s + (0.295 + 0.778i)6-s + (−0.936 + 0.350i)7-s + (0.483 − 0.0292i)8-s + (0.112 + 0.691i)9-s + (0.584 − 0.277i)10-s + (−0.0685 + 0.160i)11-s + (−0.713 − 0.101i)12-s + (−0.110 − 0.209i)13-s + (0.220 − 1.50i)14-s + (−0.225 + 0.0556i)15-s + (0.248 − 0.524i)16-s + (0.0159 + 0.0783i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0800423 - 0.172527i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0800423 - 0.172527i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.47 - 0.927i)T \) |
| 53 | \( 1 + (5.45 - 4.81i)T \) |
| good | 2 | \( 1 + (1.03 - 1.88i)T + (-1.06 - 1.69i)T^{2} \) |
| 3 | \( 1 + (-0.613 + 0.721i)T + (-0.481 - 2.96i)T^{2} \) |
| 5 | \( 1 + (0.771 + 0.555i)T + (1.58 + 4.74i)T^{2} \) |
| 11 | \( 1 + (0.227 - 0.533i)T + (-7.61 - 7.93i)T^{2} \) |
| 13 | \( 1 + (0.397 + 0.756i)T + (-7.38 + 10.6i)T^{2} \) |
| 17 | \( 1 + (-0.0659 - 0.322i)T + (-15.6 + 6.66i)T^{2} \) |
| 19 | \( 1 + (8.20 - 1.84i)T + (17.1 - 8.14i)T^{2} \) |
| 23 | \( 1 + (7.30 + 1.95i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.354 + 0.0430i)T + (28.1 + 6.94i)T^{2} \) |
| 31 | \( 1 + (5.26 - 0.747i)T + (29.7 - 8.62i)T^{2} \) |
| 37 | \( 1 + (9.09 + 4.31i)T + (23.4 + 28.6i)T^{2} \) |
| 41 | \( 1 + (-7.73 - 6.06i)T + (9.81 + 39.8i)T^{2} \) |
| 43 | \( 1 + (-0.941 - 0.649i)T + (15.2 + 40.2i)T^{2} \) |
| 47 | \( 1 + (-8.88 - 7.25i)T + (9.40 + 46.0i)T^{2} \) |
| 59 | \( 1 + (-7.22 - 1.17i)T + (55.9 + 18.6i)T^{2} \) |
| 61 | \( 1 + (-5.65 + 3.73i)T + (23.9 - 56.1i)T^{2} \) |
| 67 | \( 1 + (2.10 - 9.36i)T + (-60.5 - 28.7i)T^{2} \) |
| 71 | \( 1 + (1.17 + 6.40i)T + (-66.3 + 25.1i)T^{2} \) |
| 73 | \( 1 + (-2.27 - 1.50i)T + (28.6 + 67.1i)T^{2} \) |
| 79 | \( 1 + (0.0445 - 2.21i)T + (-78.9 - 3.18i)T^{2} \) |
| 83 | \( 1 + (4.67 - 4.67i)T - 83iT^{2} \) |
| 89 | \( 1 + (4.51 - 13.5i)T + (-71.1 - 53.4i)T^{2} \) |
| 97 | \( 1 + (10.4 + 3.95i)T + (72.6 + 64.3i)T^{2} \) |
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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
−12.38320842930646284228866779669, −10.70069482499193074065212941221, −9.858811979040727572960203039391, −8.789679089918653085336891204740, −8.209620530260338020556030827591, −7.42133039793507912605729211336, −6.45325956164259768437097714749, −5.67774953529028644069269267518, −4.17701042998935025982009820022, −2.36436655241204522304182561895,
0.14500858244389804084845687552, 2.19188878256846650768505352514, 3.52304789600147774254505285941, 3.99882468243648591865926596153, 6.07334211097693270869919107872, 7.17755327939343428738444426368, 8.559781442888770495342332064271, 9.178775063049709652755320522596, 10.04334395311260029815787505990, 10.58733318232081124316912551823
