L(s) = 1 | + (0.334 − 0.192i)2-s + (−1.42 + 0.983i)3-s + (−0.925 + 1.60i)4-s + (−0.286 + 0.603i)6-s + (−2.36 − 1.17i)7-s + 1.48i·8-s + (1.06 − 2.80i)9-s + (−2.20 − 1.27i)11-s + (−0.257 − 3.19i)12-s + 3.06i·13-s + (−1.01 + 0.0640i)14-s + (−1.56 − 2.71i)16-s + (3.23 − 5.59i)17-s + (−0.185 − 1.14i)18-s + (−1.03 + 0.597i)19-s + ⋯ |
L(s) = 1 | + (0.236 − 0.136i)2-s + (−0.823 + 0.567i)3-s + (−0.462 + 0.801i)4-s + (−0.116 + 0.246i)6-s + (−0.895 − 0.444i)7-s + 0.525i·8-s + (0.354 − 0.934i)9-s + (−0.663 − 0.383i)11-s + (−0.0743 − 0.922i)12-s + 0.850i·13-s + (−0.272 + 0.0171i)14-s + (−0.391 − 0.677i)16-s + (0.783 − 1.35i)17-s + (−0.0436 − 0.269i)18-s + (−0.237 + 0.137i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.250 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.192266 - 0.248474i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.192266 - 0.248474i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.42 - 0.983i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.36 + 1.17i)T \) |
good | 2 | \( 1 + (-0.334 + 0.192i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (2.20 + 1.27i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.06iT - 13T^{2} \) |
| 17 | \( 1 + (-3.23 + 5.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.03 - 0.597i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.64 + 1.52i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.77iT - 29T^{2} \) |
| 31 | \( 1 + (5.95 + 3.43i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.77 + 3.07i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 + 5.46T + 43T^{2} \) |
| 47 | \( 1 + (-1.61 - 2.78i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (11.4 + 6.62i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.98 - 3.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.08 - 4.67i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.75 - 3.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.921iT - 71T^{2} \) |
| 73 | \( 1 + (0.256 + 0.148i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.14 + 7.18i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.11T + 83T^{2} \) |
| 89 | \( 1 + (-9.41 - 16.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.3iT - 97T^{2} \) |
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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
−10.70556472623299110172159049757, −9.633759245378220560716400441834, −9.173383698830532410145862082961, −7.79777663246679127999123965328, −6.92270219122804458419864130892, −5.80689260149434302217527254253, −4.78377254532513269810131114644, −3.89084683499449962798064835849, −2.91386304881714665090032851260, −0.19343976633951848076943006416,
1.53961666812119611754399365543, 3.32124490301189422930415205283, 4.91248279957756894030366677779, 5.59191084464654822558120904473, 6.31184309835196455736834817932, 7.26599939272395662102237744061, 8.409267603726501923544784853567, 9.526484747413626630602497121762, 10.46050175904670522589962188182, 10.80454620197755469391894954499