| L(s) = 1 | + (−1.30 + 1.51i)2-s + (−2.52 + 1.62i)3-s + (−0.569 − 3.95i)4-s + (2.02 + 4.44i)5-s + (0.853 − 5.93i)6-s + (−16.2 + 4.77i)7-s + (6.73 + 4.32i)8-s + (3.73 − 8.18i)9-s + (−9.36 − 2.75i)10-s + (−9.92 − 11.4i)11-s + (7.85 + 9.06i)12-s + (−17.3 − 5.08i)13-s + (14.0 − 30.8i)14-s + (−12.3 − 7.91i)15-s + (−15.3 + 4.50i)16-s + (11.0 − 76.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.181 + 0.397i)5-s + (0.0580 − 0.404i)6-s + (−0.878 + 0.257i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.296 − 0.0869i)10-s + (−0.272 − 0.314i)11-s + (0.189 + 0.218i)12-s + (−0.369 − 0.108i)13-s + (0.268 − 0.588i)14-s + (−0.212 − 0.136i)15-s + (−0.239 + 0.0704i)16-s + (0.157 − 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.308430 - 0.252207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.308430 - 0.252207i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (7.59 + 110. i)T \) |
| good | 5 | \( 1 + (-2.02 - 4.44i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (16.2 - 4.77i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (9.92 + 11.4i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (17.3 + 5.08i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-11.0 + 76.9i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (0.468 + 3.26i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-31.7 + 220. i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-11.9 - 7.66i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (36.4 - 79.8i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (29.8 + 65.3i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (268. - 172. i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 + 416.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-19.8 + 5.83i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (131. + 38.7i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (482. + 310. i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (154. - 177. i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-209. + 241. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-156. - 1.08e3i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (131. + 38.4i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (231. - 506. i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-662. + 425. i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (173. + 379. i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
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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
−12.43864584857289906057827865976, −11.31231121537900088093121869175, −10.16544566760194084377662858380, −9.568005022476260747285747014946, −8.260879066530174835085718660932, −6.87093146454067600653971644030, −6.08451933909028736384057679201, −4.79063550998537060535082399172, −2.85448242471065783379541591847, −0.24696336868877298326315719567,
1.55287008849914166372993513099, 3.41826433347919124624557280836, 5.08546700343556736934373153876, 6.51894424914384813221518011600, 7.62701217476415754129400570599, 8.946172032815678595923788984218, 9.961298771303194971453955348051, 10.78294109027450405016563369704, 12.01732811279121381950022788835, 12.79682786076058954864979562233
