L(s) = 1 | + (−9 − 15.5i)5-s + (18 − 31.1i)11-s − 10·13-s + (9 − 15.5i)17-s + (50 + 86.6i)19-s + (36 + 62.3i)23-s + (−99.5 + 172. i)25-s + 234·29-s + (8 − 13.8i)31-s + (113 + 195. i)37-s − 90·41-s + 452·43-s + (216 + 374. i)47-s + (207 − 358. i)53-s − 648·55-s + ⋯ |
L(s) = 1 | + (−0.804 − 1.39i)5-s + (0.493 − 0.854i)11-s − 0.213·13-s + (0.128 − 0.222i)17-s + (0.603 + 1.04i)19-s + (0.326 + 0.565i)23-s + (−0.796 + 1.37i)25-s + 1.49·29-s + (0.0463 − 0.0802i)31-s + (0.502 + 0.869i)37-s − 0.342·41-s + 1.60·43-s + (0.670 + 1.16i)47-s + (0.536 − 0.929i)53-s − 1.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.926060383\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.926060383\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (9 + 15.5i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-18 + 31.1i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 10T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-9 + 15.5i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-50 - 86.6i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-36 - 62.3i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 234T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-8 + 13.8i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-113 - 195. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 90T + 6.89e4T^{2} \) |
| 43 | \( 1 - 452T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-216 - 374. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-207 + 358. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (342 - 592. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (211 + 365. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (166 - 287. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 360T + 3.57e5T^{2} \) |
| 73 | \( 1 + (13 - 22.5i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (256 + 443. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (315 + 545. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−8.810075573587609535778008108965, −8.056093602934370311458087087411, −7.52798203144024547504946464917, −6.30860509886872680089613247630, −5.48842906634384018100220404469, −4.65806581260244341721092494583, −3.91211950701074166263813684002, −2.96498771879852714710953592301, −1.32217538116967543484388056472, −0.69265111181377149649628938626,
0.71064849591796694813613207809, 2.31339744177336011169806873656, 3.03543768132552600439369260383, 4.02541703105511403385011242021, 4.77525633506588173175191203784, 6.03717887113895855828917687679, 6.96250339490282058935629204592, 7.20882021205489491860324162914, 8.119459920017407908967206283015, 9.100784485014251507452589711836