L(s) = 1 | + (−4.65 − 2.31i)3-s − 0.943·5-s + 7i·7-s + (16.2 + 21.5i)9-s − 17.0i·11-s − 27.1i·13-s + (4.38 + 2.18i)15-s − 6.69i·17-s − 24.6·19-s + (16.2 − 32.5i)21-s + 67.7·23-s − 124.·25-s + (−25.5 − 137. i)27-s − 112.·29-s + 223. i·31-s + ⋯ |
L(s) = 1 | + (−0.894 − 0.446i)3-s − 0.0843·5-s + 0.377i·7-s + (0.601 + 0.798i)9-s − 0.467i·11-s − 0.579i·13-s + (0.0755 + 0.0376i)15-s − 0.0954i·17-s − 0.297·19-s + (0.168 − 0.338i)21-s + 0.614·23-s − 0.992·25-s + (−0.182 − 0.983i)27-s − 0.717·29-s + 1.29i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9347786127\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9347786127\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.65 + 2.31i)T \) |
| 7 | \( 1 - 7iT \) |
good | 5 | \( 1 + 0.943T + 125T^{2} \) |
| 11 | \( 1 + 17.0iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 27.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 6.69iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 24.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 67.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 112.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 223. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 260. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 164. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 96.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 8.05T + 1.03e5T^{2} \) |
| 53 | \( 1 - 122.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 750. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 145. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 843.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 816.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 657.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 624. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 358. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 721.T + 9.12e5T^{2} \) |
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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
−10.47148733553468655161577488398, −9.430043535319250309078582690060, −8.386935091657773517924946655819, −7.53732773790203505418266198131, −6.61489228310930347311204528281, −5.71177380127984237536806347870, −5.03448406719807255768910133800, −3.69998379412825674750332135264, −2.28686187077110599341207786932, −0.920707523651024469322195219536,
0.38065692185266280827016252918, 1.87765401777206190906509852561, 3.63331745260526508985316925041, 4.43996429129822082302507255006, 5.37863830440353082209892859999, 6.40565310874438123881628167019, 7.13857736400433237637721836048, 8.200157001523525006978540345814, 9.480429076581513597649478006120, 9.887142693557850069870607724367