L(s) = 1 | + 1.41·2-s + (−2.88 + 0.821i)3-s + 2.00·4-s + (−4.52 − 2.12i)5-s + (−4.08 + 1.16i)6-s + 2.64i·7-s + 2.82·8-s + (7.64 − 4.74i)9-s + (−6.39 − 3.01i)10-s − 18.1i·11-s + (−5.77 + 1.64i)12-s − 22.5i·13-s + 3.74i·14-s + (14.8 + 2.42i)15-s + 4.00·16-s − 0.457·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.961 + 0.273i)3-s + 0.500·4-s + (−0.904 − 0.425i)5-s + (−0.680 + 0.193i)6-s + 0.377i·7-s + 0.353·8-s + (0.849 − 0.526i)9-s + (−0.639 − 0.301i)10-s − 1.65i·11-s + (−0.480 + 0.136i)12-s − 1.73i·13-s + 0.267i·14-s + (0.986 + 0.161i)15-s + 0.250·16-s − 0.0268·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.687738 - 0.809590i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.687738 - 0.809590i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 + (2.88 - 0.821i)T \) |
| 5 | \( 1 + (4.52 + 2.12i)T \) |
| 7 | \( 1 - 2.64iT \) |
good | 11 | \( 1 + 18.1iT - 121T^{2} \) |
| 13 | \( 1 + 22.5iT - 169T^{2} \) |
| 17 | \( 1 + 0.457T + 289T^{2} \) |
| 19 | \( 1 + 30.1T + 361T^{2} \) |
| 23 | \( 1 + 12.3T + 529T^{2} \) |
| 29 | \( 1 - 4.70iT - 841T^{2} \) |
| 31 | \( 1 - 45.2T + 961T^{2} \) |
| 37 | \( 1 + 32.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 22.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 20.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 9.67T + 2.20e3T^{2} \) |
| 53 | \( 1 - 5.97T + 2.80e3T^{2} \) |
| 59 | \( 1 - 112. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 56.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 67.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 20.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 97.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 41.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 121.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 26.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 151. iT - 9.40e3T^{2} \) |
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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
−11.98804641557178829922382261561, −11.00352543740748505921098458007, −10.46059322566017172240079665613, −8.711698504554694287233257205199, −7.81357930436088558730965532003, −6.26341522464514607306533284999, −5.55597336284482443852775730962, −4.39492583177297239867641993061, −3.22764070050728178874387865328, −0.52873235373171506438856168794,
2.00539133192956337367488274056, 4.24297499033257243814920384141, 4.59183992241477021789717465499, 6.54539889626060218055027621957, 6.85316109816884291570658361430, 8.024274458388316275848873312568, 9.847914115473493575306493672656, 10.76624941596667811832402263676, 11.73652466137786492977585596662, 12.17803564361464903896319018437