| 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
−12.79682786076058954864979562233, −12.01732811279121381950022788835, −10.78294109027450405016563369704, −9.961298771303194971453955348051, −8.946172032815678595923788984218, −7.62701217476415754129400570599, −6.51894424914384813221518011600, −5.08546700343556736934373153876, −3.41826433347919124624557280836, −1.55287008849914166372993513099,
0.24696336868877298326315719567, 2.85448242471065783379541591847, 4.79063550998537060535082399172, 6.08451933909028736384057679201, 6.87093146454067600653971644030, 8.260879066530174835085718660932, 9.568005022476260747285747014946, 10.16544566760194084377662858380, 11.31231121537900088093121869175, 12.43864584857289906057827865976
