| L(s) = 1 | + (122. + 212. i)3-s + (−1.69e3 − 977. i)5-s + (3.57e3 + 5.25e3i)7-s + (−2.03e4 + 3.52e4i)9-s + (−4.58e4 + 2.64e4i)11-s + 1.92e5i·13-s − 4.80e5i·15-s + (5.05e5 − 2.91e5i)17-s + (−4.29e4 + 7.43e4i)19-s + (−6.78e5 + 1.40e6i)21-s + (−7.78e5 − 4.49e5i)23-s + (9.33e5 + 1.61e6i)25-s − 5.17e6·27-s − 1.30e6·29-s + (−8.22e5 − 1.42e6i)31-s + ⋯ |
| L(s) = 1 | + (0.876 + 1.51i)3-s + (−1.21 − 0.699i)5-s + (0.562 + 0.826i)7-s + (−1.03 + 1.79i)9-s + (−0.943 + 0.544i)11-s + 1.87i·13-s − 2.45i·15-s + (1.46 − 0.847i)17-s + (−0.0756 + 0.130i)19-s + (−0.761 + 1.57i)21-s + (−0.579 − 0.334i)23-s + (0.478 + 0.828i)25-s − 1.87·27-s − 0.341·29-s + (−0.159 − 0.276i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9263366551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9263366551\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-3.57e3 - 5.25e3i)T \) |
| good | 3 | \( 1 + (-122. - 212. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (1.69e3 + 977. i)T + (9.76e5 + 1.69e6i)T^{2} \) |
| 11 | \( 1 + (4.58e4 - 2.64e4i)T + (1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 - 1.92e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 + (-5.05e5 + 2.91e5i)T + (5.92e10 - 1.02e11i)T^{2} \) |
| 19 | \( 1 + (4.29e4 - 7.43e4i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (7.78e5 + 4.49e5i)T + (9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + 1.30e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (8.22e5 + 1.42e6i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (8.73e6 - 1.51e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + 1.29e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + 3.51e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + (-5.60e5 + 9.71e5i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (3.21e7 + 5.56e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-2.74e7 - 4.74e7i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (1.77e7 + 1.02e7i)T + (5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-5.78e5 + 3.33e5i)T + (1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.38e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.11e8 + 6.45e7i)T + (2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (5.11e8 + 2.95e8i)T + (5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + 2.21e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-8.81e8 - 5.08e8i)T + (1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + 3.81e8iT - 7.60e17T^{2} \) |
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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
−12.17716513174151557895169507492, −11.57594054762505131193292568319, −10.21902494556418655269899466521, −9.214496429515011898289329557906, −8.468740537891902103885230396672, −7.58811012981705078540454375248, −5.18388353264028806941891540923, −4.52638897477541723618341547556, −3.51105939115829139119479247786, −2.10177208354834541939399869100,
0.21633435581815496503535362449, 1.24057338646621314159755003292, 2.89684684858901846944001994030, 3.59207868130104080215723977152, 5.73773517971552103543874422552, 7.26975735949965885383024142117, 7.943539012569000496930994212735, 8.128726394439070775907609836335, 10.31943780574609051468635817263, 11.23554006111873603394668381100
