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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.561073825\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.561073825\) |
\(L(\frac{5}{2})\) |
|
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
−10.49168572542766451764334176366, −9.763672518757408595995853031112, −9.033637644020368249720404255201, −7.54961087499234549795456144267, −6.71024968894050877065687948093, −5.82149862418698586655244389073, −4.61248311365206293697860066190, −2.77966791934384662347768022842, −2.31807257929985888782516018828, −0.60849212234172808084936253559,
1.22950439247444939125282418481, 2.83951920831526033577940649892, 4.13046858263578429783260201544, 5.65821502973934258610422534175, 6.09514495664112271851599783853, 7.44646408232901722560969183137, 8.417487788161749205090133288766, 8.701428790699868430407146015810, 9.966885573262831971676675036720, 10.80007636674251528990889735361