| L(s) = 1 | + (1.30 − 1.51i)2-s + (2.52 − 1.62i)3-s + (−0.569 − 3.95i)4-s + (5.61 + 12.2i)5-s + (0.853 − 5.93i)6-s + (18.2 − 5.37i)7-s + (−6.73 − 4.32i)8-s + (3.73 − 8.18i)9-s + (25.9 + 7.61i)10-s + (1.50 + 1.73i)11-s + (−7.85 − 9.06i)12-s + (42.5 + 12.5i)13-s + (15.8 − 34.6i)14-s + (34.1 + 21.9i)15-s + (−15.3 + 4.50i)16-s + (1.53 − 10.6i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.502 + 1.09i)5-s + (0.0580 − 0.404i)6-s + (0.988 − 0.290i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (0.819 + 0.240i)10-s + (0.0413 + 0.0476i)11-s + (−0.189 − 0.218i)12-s + (0.908 + 0.266i)13-s + (0.302 − 0.662i)14-s + (0.586 + 0.377i)15-s + (−0.239 + 0.0704i)16-s + (0.0218 − 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.64619 - 1.01413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.64619 - 1.01413i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 + 1.51i)T \) |
| 3 | \( 1 + (-2.52 + 1.62i)T \) |
| 23 | \( 1 + (71.7 + 83.8i)T \) |
| good | 5 | \( 1 + (-5.61 - 12.2i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (-18.2 + 5.37i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (-1.50 - 1.73i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-42.5 - 12.5i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-1.53 + 10.6i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (5.62 + 39.0i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (18.2 - 126. i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (178. + 114. i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (14.5 - 31.7i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-206. - 451. i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-68.6 + 44.1i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 + 193.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (416. - 122. i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-414. - 121. i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (316. + 203. i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (243. - 280. i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (685. - 791. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-11.0 - 76.7i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-348. - 102. i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-305. + 668. i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (1.11e3 - 713. i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (655. + 1.43e3i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
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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
−12.75544389165127222200828512509, −11.35354329568397504277750551741, −10.84047760542859851875131470494, −9.676475263767691033976285842292, −8.379485157496138903437139651543, −7.09004962451544149811421604967, −6.00316937038871392242142015167, −4.36089958676548978550434142100, −2.92733087438701668966164518714, −1.63252142671726661952323197883,
1.72883710456997338766691967462, 3.80833721891079996915936224334, 5.04451907248729734844751217058, 5.89616548207142069590954692818, 7.73538069581415738325007732031, 8.549358616589029214626151940671, 9.366914369403925795407655419061, 10.84090196229637865464926818079, 12.07206942387493820569639671895, 13.06219450666795919019164695941
