L(s) = 1 | + (−0.979 + 1.69i)2-s + (2.07 + 3.60i)4-s + (−5.94 + 10.2i)5-s − 23.8·8-s + (−11.6 − 20.1i)10-s + (−18.2 − 31.5i)11-s + 0.964·13-s + (6.71 − 11.6i)16-s + (−49.1 − 85.0i)17-s + (53.0 − 91.7i)19-s − 49.4·20-s + 71.3·22-s + (−27.1 + 47.0i)23-s + (−8.18 − 14.1i)25-s + (−0.945 + 1.63i)26-s + ⋯ |
L(s) = 1 | + (−0.346 + 0.600i)2-s + (0.259 + 0.450i)4-s + (−0.531 + 0.920i)5-s − 1.05·8-s + (−0.368 − 0.638i)10-s + (−0.499 − 0.864i)11-s + 0.0205·13-s + (0.104 − 0.181i)16-s + (−0.700 − 1.21i)17-s + (0.639 − 1.10i)19-s − 0.552·20-s + 0.691·22-s + (−0.246 + 0.426i)23-s + (−0.0654 − 0.113i)25-s + (−0.00712 + 0.0123i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6429521813\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6429521813\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.979 - 1.69i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (5.94 - 10.2i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (18.2 + 31.5i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 0.964T + 2.19e3T^{2} \) |
| 17 | \( 1 + (49.1 + 85.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-53.0 + 91.7i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (27.1 - 47.0i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (63.8 + 110. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (155. - 270. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 419.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 523.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (135. - 234. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-125. - 217. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (204. + 353. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-430. + 745. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (253. + 438. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 523.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (314. + 545. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (159. - 276. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-174. + 301. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 161.T + 9.12e5T^{2} \) |
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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
−10.82932277869172919016852302581, −9.522247027457222342252035661174, −8.617916824546585510423677612731, −7.71157963973204052753323815582, −7.03065614490013557282602486467, −6.27322644281921561242734490511, −4.92587762886014198346186820828, −3.32418571185271811439654864441, −2.72080938537479093680353264901, −0.25238985812624729445706779386,
1.16192226024132604483362349050, 2.27942219356747329938476584343, 3.84053411470206319594038643470, 4.95978182765986758690151417410, 5.99111767320038435442227734389, 7.16758876933482901289097038230, 8.365823133822719350738512418088, 8.949208303148849141390740319835, 10.24588583463618973926857526254, 10.46017155252778232310195801678