| L(s) = 1 | + (−1.05 + 0.944i)2-s + (−0.108 − 0.129i)3-s + (0.214 − 1.98i)4-s + (2.23 − 0.0637i)5-s + (0.236 + 0.0335i)6-s + (0.176 − 0.305i)7-s + (1.65 + 2.29i)8-s + (0.515 − 2.92i)9-s + (−2.29 + 2.17i)10-s + (0.465 − 0.268i)11-s + (−0.280 + 0.188i)12-s + (−2.90 − 2.44i)13-s + (0.102 + 0.487i)14-s + (−0.250 − 0.282i)15-s + (−3.90 − 0.853i)16-s + (−1.16 + 0.204i)17-s + ⋯ |
| L(s) = 1 | + (−0.744 + 0.668i)2-s + (−0.0626 − 0.0747i)3-s + (0.107 − 0.994i)4-s + (0.999 − 0.0285i)5-s + (0.0965 + 0.0137i)6-s + (0.0666 − 0.115i)7-s + (0.584 + 0.811i)8-s + (0.171 − 0.975i)9-s + (−0.724 + 0.689i)10-s + (0.140 − 0.0810i)11-s + (−0.0810 + 0.0543i)12-s + (−0.807 − 0.677i)13-s + (0.0275 + 0.130i)14-s + (−0.0647 − 0.0728i)15-s + (−0.976 − 0.213i)16-s + (−0.281 + 0.0497i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06466 - 0.133956i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06466 - 0.133956i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.05 - 0.944i)T \) |
| 5 | \( 1 + (-2.23 + 0.0637i)T \) |
| 19 | \( 1 + (-1.57 + 4.06i)T \) |
| good | 3 | \( 1 + (0.108 + 0.129i)T + (-0.520 + 2.95i)T^{2} \) |
| 7 | \( 1 + (-0.176 + 0.305i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.465 + 0.268i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.90 + 2.44i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.16 - 0.204i)T + (15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-5.38 + 1.96i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-2.19 - 0.386i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.30 - 5.72i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.773T + 37T^{2} \) |
| 41 | \( 1 + (2.78 + 3.31i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-10.6 - 3.87i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.460 + 2.61i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.14 + 1.50i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.583 - 3.30i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (6.49 - 2.36i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (6.98 + 1.23i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.72 - 2.44i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (7.32 + 8.73i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (5.66 - 4.74i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.764 - 1.32i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.25 + 3.87i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.09 - 11.8i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.94361959640127977788160083528, −10.24525748110330479169009423406, −9.257269818615531818303923738768, −8.858415127723442745211286631141, −7.37435922362448229808229243710, −6.71101463649477022896701474481, −5.72302641554471890030815045237, −4.77546179077975774135928992271, −2.71925023179925203318487628360, −1.01794927797318014274061574163,
1.68360296898390493650252337995, 2.67195888785119196825511429882, 4.33661074501048423009103606449, 5.51256829725389058328194424378, 6.91075688127224464024420019234, 7.75266124090674981533597646870, 8.950673609154885131789868133110, 9.629306203514087429950533955539, 10.39283295775462103712815828028, 11.18866482617235208841750743462
