| L(s) = 1 | + (−1.10 − 5.22i)2-s + (−1.99 − 4.79i)3-s + (−18.7 + 8.33i)4-s + (−7.33 + 8.43i)5-s + (−22.8 + 15.7i)6-s + (18.8 − 10.8i)7-s + (39.1 + 53.9i)8-s + (−19.0 + 19.1i)9-s + (52.1 + 28.9i)10-s + (11.8 − 2.50i)11-s + (77.3 + 73.1i)12-s + (−4.12 + 19.4i)13-s + (−77.7 − 86.3i)14-s + (55.1 + 18.3i)15-s + (128. − 142. i)16-s + (31.5 + 43.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.392 − 1.84i)2-s + (−0.384 − 0.923i)3-s + (−2.34 + 1.04i)4-s + (−0.656 + 0.754i)5-s + (−1.55 + 1.07i)6-s + (1.01 − 0.587i)7-s + (1.73 + 2.38i)8-s + (−0.704 + 0.709i)9-s + (1.65 + 0.914i)10-s + (0.323 − 0.0687i)11-s + (1.86 + 1.76i)12-s + (−0.0881 + 0.414i)13-s + (−1.48 − 1.64i)14-s + (0.948 + 0.315i)15-s + (2.00 − 2.22i)16-s + (0.450 + 0.619i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.201 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.601317 - 0.490445i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.601317 - 0.490445i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.99 + 4.79i)T \) |
| 5 | \( 1 + (7.33 - 8.43i)T \) |
| good | 2 | \( 1 + (1.10 + 5.22i)T + (-7.30 + 3.25i)T^{2} \) |
| 7 | \( 1 + (-18.8 + 10.8i)T + (171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-11.8 + 2.50i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (4.12 - 19.4i)T + (-2.00e3 - 893. i)T^{2} \) |
| 17 | \( 1 + (-31.5 - 43.4i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (108. - 78.5i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-91.5 + 82.4i)T + (1.27e3 - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (-5.32 + 50.6i)T + (-2.38e4 - 5.07e3i)T^{2} \) |
| 31 | \( 1 + (-26.0 - 248. i)T + (-2.91e4 + 6.19e3i)T^{2} \) |
| 37 | \( 1 + (177. - 57.6i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-384. - 81.7i)T + (6.29e4 + 2.80e4i)T^{2} \) |
| 43 | \( 1 + (444. - 256. i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-90.3 - 9.49i)T + (1.01e5 + 2.15e4i)T^{2} \) |
| 53 | \( 1 + (-246. + 339. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (199. + 42.4i)T + (1.87e5 + 8.35e4i)T^{2} \) |
| 61 | \( 1 + (-807. + 171. i)T + (2.07e5 - 9.23e4i)T^{2} \) |
| 67 | \( 1 + (-314. + 33.0i)T + (2.94e5 - 6.25e4i)T^{2} \) |
| 71 | \( 1 + (-204. - 148. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-463. - 150. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (60.9 - 579. i)T + (-4.82e5 - 1.02e5i)T^{2} \) |
| 83 | \( 1 + (-307. + 691. i)T + (-3.82e5 - 4.24e5i)T^{2} \) |
| 89 | \( 1 + (192. - 592. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-370. - 38.9i)T + (8.92e5 + 1.89e5i)T^{2} \) |
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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
−11.46685804150634331319344695414, −10.86451335244713738008077992203, −10.23723999566281367284430834656, −8.508656840505023928087101303988, −8.019511480063098703364262829543, −6.70076505875454550471398993169, −4.75261377566837005246078403198, −3.63564950682367477831583059579, −2.19290457118791963257249172855, −1.07924903547712470720031597672,
0.49937667635846207099288132032, 4.13310841672314603909868422648, 5.02795572736567546933871002007, 5.59860373588437871460433308932, 7.02642803346863720503558467361, 8.143288076642305784313236411532, 8.836081305901393761352658346440, 9.497110638087753877916423200738, 10.89942166247588519831015224076, 11.93781896748814649070408183845
