L(s) = 1 | + (−4.48 + 3.45i)2-s + (−10.3 + 11.6i)3-s + (8.18 − 30.9i)4-s + 25i·5-s + (6.29 − 87.9i)6-s − 180. i·7-s + (70.0 + 166. i)8-s + (−28.0 − 241. i)9-s + (−86.2 − 112. i)10-s − 274.·11-s + (275. + 415. i)12-s + 983.·13-s + (622. + 808. i)14-s + (−291. − 259. i)15-s + (−890. − 506. i)16-s + 225. i·17-s + ⋯ |
L(s) = 1 | + (−0.792 + 0.610i)2-s + (−0.664 + 0.746i)3-s + (0.255 − 0.966i)4-s + 0.447i·5-s + (0.0713 − 0.997i)6-s − 1.39i·7-s + (0.387 + 0.922i)8-s + (−0.115 − 0.993i)9-s + (−0.272 − 0.354i)10-s − 0.682·11-s + (0.551 + 0.833i)12-s + 1.61·13-s + (0.848 + 1.10i)14-s + (−0.334 − 0.297i)15-s + (−0.869 − 0.494i)16-s + 0.189i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.551i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.833 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.846770 + 0.254847i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.846770 + 0.254847i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (4.48 - 3.45i)T \) |
| 3 | \( 1 + (10.3 - 11.6i)T \) |
| 5 | \( 1 - 25iT \) |
good | 7 | \( 1 + 180. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 274.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 983.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 225. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.11e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 3.88e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.81e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 4.99e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 6.76e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.47e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 8.63e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 5.28e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.41e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.29e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.06e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 642. iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 3.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.90e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 4.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.35e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 4.78e4T + 8.58e9T^{2} \) |
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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
−14.52233477405834906722578273002, −13.32147004798241503221608430917, −11.19377699887943809544596147372, −10.66651661599006189578830239899, −9.747240764571943123813204082128, −8.181344824991740159880434761010, −6.83693015075532022397509421160, −5.67585713247546225842554582825, −3.92595263415971068345234938966, −0.814402507673302769976400741834,
1.03316373992490499612791284103, 2.66389379841854783701990119001, 5.23503017360918355360419735940, 6.74086471664789793382787763168, 8.305226434065815516732796696673, 9.072662877633570725206215881145, 10.86614164287504446572822625273, 11.56631730505387910968186004269, 12.73359029843581982631664094976, 13.30062180949272706966832402719