L(s) = 1 | − 181. i·2-s + (6.17e3 + 2.20e3i)3-s − 3.27e4·4-s + 5.48e5i·5-s + (3.99e5 − 1.11e6i)6-s − 8.81e6·7-s + 5.93e6i·8-s + (3.32e7 + 2.72e7i)9-s + 9.92e7·10-s + 2.97e8i·11-s + (−2.02e8 − 7.23e7i)12-s + 6.77e8·13-s + 1.59e9i·14-s + (−1.21e9 + 3.38e9i)15-s + 1.07e9·16-s − 3.75e9i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.941 + 0.336i)3-s − 0.500·4-s + 1.40i·5-s + (0.238 − 0.665i)6-s − 1.52·7-s + 0.353i·8-s + (0.773 + 0.634i)9-s + 0.992·10-s + 1.38i·11-s + (−0.470 − 0.168i)12-s + 0.830·13-s + 1.08i·14-s + (−0.472 + 1.32i)15-s + 0.250·16-s − 0.538i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{17}{2})\) |
\(\approx\) |
\(1.38795 + 0.977760i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38795 + 0.977760i\) |
\(L(9)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 181. iT \) |
| 3 | \( 1 + (-6.17e3 - 2.20e3i)T \) |
good | 5 | \( 1 - 5.48e5iT - 1.52e11T^{2} \) |
| 7 | \( 1 + 8.81e6T + 3.32e13T^{2} \) |
| 11 | \( 1 - 2.97e8iT - 4.59e16T^{2} \) |
| 13 | \( 1 - 6.77e8T + 6.65e17T^{2} \) |
| 17 | \( 1 + 3.75e9iT - 4.86e19T^{2} \) |
| 19 | \( 1 + 9.70e8T + 2.88e20T^{2} \) |
| 23 | \( 1 + 2.68e10iT - 6.13e21T^{2} \) |
| 29 | \( 1 - 3.75e11iT - 2.50e23T^{2} \) |
| 31 | \( 1 + 4.78e11T + 7.27e23T^{2} \) |
| 37 | \( 1 - 9.79e11T + 1.23e25T^{2} \) |
| 41 | \( 1 + 1.07e13iT - 6.37e25T^{2} \) |
| 43 | \( 1 - 8.94e12T + 1.36e26T^{2} \) |
| 47 | \( 1 - 2.80e13iT - 5.66e26T^{2} \) |
| 53 | \( 1 + 7.34e13iT - 3.87e27T^{2} \) |
| 59 | \( 1 - 1.35e14iT - 2.15e28T^{2} \) |
| 61 | \( 1 - 4.44e13T + 3.67e28T^{2} \) |
| 67 | \( 1 - 6.70e14T + 1.64e29T^{2} \) |
| 71 | \( 1 - 6.75e14iT - 4.16e29T^{2} \) |
| 73 | \( 1 - 4.82e14T + 6.50e29T^{2} \) |
| 79 | \( 1 - 3.05e14T + 2.30e30T^{2} \) |
| 83 | \( 1 + 1.62e15iT - 5.07e30T^{2} \) |
| 89 | \( 1 - 3.89e15iT - 1.54e31T^{2} \) |
| 97 | \( 1 - 1.41e16T + 6.14e31T^{2} \) |
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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
−19.32878555191323513607667324831, −18.32050377136498189440019602331, −15.74917290751665689999918988256, −14.35699216036824260266236093852, −12.86408880357099527148798080929, −10.53725329518493237302244383137, −9.450931552343509865561183915468, −6.99688512224407447495996652940, −3.66337044170614071331562990011, −2.49031371516011271071657554530,
0.74879565564076440136694501021, 3.63165854064562074591928381867, 6.13930717651991674205615488518, 8.316903820859656618193592963459, 9.353985126093516907070868432293, 12.81505850937525346743821105561, 13.59127463853632432735875182465, 15.73428745662231062758418438088, 16.64234089813135749250176796230, 18.82310474221468497564362417951