L(s) = 1 | + (0.331 − 7.99i)2-s − 15.5i·3-s + (−63.7 − 5.30i)4-s − 55.9·5-s + (−124. − 5.17i)6-s − 552. i·7-s + (−63.5 + 508. i)8-s − 243·9-s + (−18.5 + 446. i)10-s + 1.28e3i·11-s + (−82.7 + 994. i)12-s − 2.51e3·13-s + (−4.41e3 − 183. i)14-s + 871. i·15-s + (4.03e3 + 676. i)16-s + 4.59e3·17-s + ⋯ |
L(s) = 1 | + (0.0414 − 0.999i)2-s − 0.577i·3-s + (−0.996 − 0.0829i)4-s − 0.447·5-s + (−0.576 − 0.0239i)6-s − 1.60i·7-s + (−0.124 + 0.992i)8-s − 0.333·9-s + (−0.0185 + 0.446i)10-s + 0.965i·11-s + (−0.0478 + 0.575i)12-s − 1.14·13-s + (−1.60 − 0.0667i)14-s + 0.258i·15-s + (0.986 + 0.165i)16-s + 0.935·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0829 - 0.996i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.0829 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.202175 + 0.186050i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.202175 + 0.186050i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.331 + 7.99i)T \) |
| 3 | \( 1 + 15.5iT \) |
| 5 | \( 1 + 55.9T \) |
good | 7 | \( 1 + 552. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 1.28e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 2.51e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 4.59e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 9.80e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.52e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.43e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 3.44e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 7.14e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 3.76e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.53e3iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.31e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.28e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 2.71e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.52e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 7.17e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 1.93e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.25e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 6.09e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 5.92e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.54e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 1.46e6T + 8.32e11T^{2} \) |
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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
−12.63053118285361916995357548196, −12.12387861241103739028595849423, −10.59533640065395326789000629763, −9.906405141601502351058193671616, −8.079823345184872981663215679080, −7.09524194516886329495255272518, −4.84387452155881615377696602247, −3.53420984483801053073431435718, −1.61493267711635289180160181245, −0.11412494824800675948380878915,
3.14633926545379518070777509663, 4.97051804345147032861270544935, 5.87929315886679713054378730639, 7.59818007795872532832633449267, 8.819914662151858183981743986263, 9.599638919967430150913153959019, 11.43552407097029540403575338866, 12.50079640466554993007777448230, 13.92586504467722143640630154206, 15.14759099950926607933131547075