L(s) = 1 | + i·2-s + (−1.68 + 0.403i)3-s − 4-s + (−0.403 − 1.68i)6-s + (−2.31 + 1.28i)7-s − i·8-s + (2.67 − 1.35i)9-s + 5.34i·11-s + (1.68 − 0.403i)12-s − 3.95i·13-s + (−1.28 − 2.31i)14-s + 16-s − 7.32·17-s + (1.35 + 2.67i)18-s + 0.807i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.972 + 0.232i)3-s − 0.5·4-s + (−0.164 − 0.687i)6-s + (−0.874 + 0.484i)7-s − 0.353i·8-s + (0.891 − 0.453i)9-s + 1.61i·11-s + (0.486 − 0.116i)12-s − 1.09i·13-s + (−0.342 − 0.618i)14-s + 0.250·16-s − 1.77·17-s + (0.320 + 0.630i)18-s + 0.185i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3411451917\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3411451917\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (1.68 - 0.403i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.31 - 1.28i)T \) |
good | 11 | \( 1 - 5.34iT - 11T^{2} \) |
| 13 | \( 1 + 3.95iT - 13T^{2} \) |
| 17 | \( 1 + 7.32T + 17T^{2} \) |
| 19 | \( 1 - 0.807iT - 19T^{2} \) |
| 23 | \( 1 + 0.281iT - 23T^{2} \) |
| 29 | \( 1 + 0.281iT - 29T^{2} \) |
| 31 | \( 1 + 9.07iT - 31T^{2} \) |
| 37 | \( 1 - 6.06T + 37T^{2} \) |
| 41 | \( 1 + 6.15T + 41T^{2} \) |
| 43 | \( 1 - 6.34T + 43T^{2} \) |
| 47 | \( 1 - 5.78T + 47T^{2} \) |
| 53 | \( 1 + 10.9iT - 53T^{2} \) |
| 59 | \( 1 - 4.90T + 59T^{2} \) |
| 61 | \( 1 + 13.2iT - 61T^{2} \) |
| 67 | \( 1 + 6.71T + 67T^{2} \) |
| 71 | \( 1 + 3.36iT - 71T^{2} \) |
| 73 | \( 1 - 4.98iT - 73T^{2} \) |
| 79 | \( 1 + 3.26T + 79T^{2} \) |
| 83 | \( 1 - 1.53T + 83T^{2} \) |
| 89 | \( 1 - 4.31T + 89T^{2} \) |
| 97 | \( 1 - 15.0iT - 97T^{2} \) |
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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
−9.734139839663403187832256306593, −9.191700137400871395004750432280, −7.964834957060348081954651790464, −7.03090360719705128488188985460, −6.42555952565073363198174032958, −5.62625544543816966118978708191, −4.73292835751712898114803004619, −3.92685046050120773943335277086, −2.29685301149355771755722748684, −0.20222678518793418210958281516,
1.10115525474682899880370079157, 2.61841997007074116825203943452, 3.86742693588533508918653570831, 4.63461774457505262173126808118, 5.86593486433773230876238051548, 6.52473388509827624283045884607, 7.29776420317175824181057322759, 8.719798572341845104316181932844, 9.197025479264362243770763892984, 10.39175290015866573581120027654