L(s) = 1 | + 5.18i·2-s + 3i·3-s − 18.9·4-s + (−4.24 − 10.3i)5-s − 15.5·6-s + 7i·7-s − 56.5i·8-s − 9·9-s + (53.6 − 22.0i)10-s − 35.9·11-s − 56.7i·12-s − 45.2i·13-s − 36.3·14-s + (31.0 − 12.7i)15-s + 142.·16-s + 113. i·17-s + ⋯ |
L(s) = 1 | + 1.83i·2-s + 0.577i·3-s − 2.36·4-s + (−0.379 − 0.925i)5-s − 1.05·6-s + 0.377i·7-s − 2.49i·8-s − 0.333·9-s + (1.69 − 0.695i)10-s − 0.985·11-s − 1.36i·12-s − 0.965i·13-s − 0.693·14-s + (0.534 − 0.219i)15-s + 2.21·16-s + 1.61i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.379 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.207912 - 0.139437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.207912 - 0.139437i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3iT \) |
| 5 | \( 1 + (4.24 + 10.3i)T \) |
| 7 | \( 1 - 7iT \) |
good | 2 | \( 1 - 5.18iT - 8T^{2} \) |
| 11 | \( 1 + 35.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 113. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 61.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 30.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 164.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 410. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 309.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 29.9iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 483. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 295. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 151.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 89.5iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 714.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 297. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 90.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 492. iT - 9.12e5T^{2} \) |
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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
−14.61117206978981457499800122940, −13.19114425501968282548832035258, −12.61877681590840568800751756140, −10.65673932950181751339238282581, −9.335638264434699905763048422962, −8.351162358393976889089994031099, −7.76918477268921503445986724811, −5.97217843661931026807607552208, −5.22640840953907941668340536240, −4.02165309549407933467306641639,
0.13608906542958302754975368007, 2.14837348286011675999983431706, 3.28246318049781549774943209161, 4.77370533576422905793775935756, 6.87202622241605779877537475179, 8.214355049958738348785633582155, 9.620510626423757557516634153616, 10.58368085902245432827294698463, 11.44918798658959091150835948609, 12.08930858452417692122269517174