| L(s) = 1 | + (1.30 + 1.51i)2-s + (−2.52 − 1.62i)3-s + (−0.569 + 3.95i)4-s + (3.30 − 7.24i)5-s + (−0.853 − 5.93i)6-s + (−0.555 − 0.163i)7-s + (−6.73 + 4.32i)8-s + (3.73 + 8.18i)9-s + (15.2 − 4.48i)10-s + (40.4 − 46.7i)11-s + (7.85 − 9.06i)12-s + (43.6 − 12.8i)13-s + (−0.481 − 1.05i)14-s + (−20.1 + 12.9i)15-s + (−15.3 − 4.50i)16-s + (−2.00 − 13.9i)17-s + ⋯ |
| L(s) = 1 | + (0.463 + 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.295 − 0.647i)5-s + (−0.0580 − 0.404i)6-s + (−0.0300 − 0.00881i)7-s + (−0.297 + 0.191i)8-s + (0.138 + 0.303i)9-s + (0.483 − 0.141i)10-s + (1.10 − 1.28i)11-s + (0.189 − 0.218i)12-s + (0.931 − 0.273i)13-s + (−0.00918 − 0.0201i)14-s + (−0.345 + 0.222i)15-s + (−0.239 − 0.0704i)16-s + (−0.0286 − 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.90637 - 0.320804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90637 - 0.320804i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-108. - 22.1i)T \) |
| good | 5 | \( 1 + (-3.30 + 7.24i)T + (-81.8 - 94.4i)T^{2} \) |
| 7 | \( 1 + (0.555 + 0.163i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (-40.4 + 46.7i)T + (-189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-43.6 + 12.8i)T + (1.84e3 - 1.18e3i)T^{2} \) |
| 17 | \( 1 + (2.00 + 13.9i)T + (-4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-9.69 + 67.4i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (2.17 + 15.1i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (50.9 - 32.7i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (131. + 288. i)T + (-3.31e4 + 3.82e4i)T^{2} \) |
| 41 | \( 1 + (202. - 444. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (91.0 + 58.5i)T + (3.30e4 + 7.23e4i)T^{2} \) |
| 47 | \( 1 - 170.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (57.9 + 17.0i)T + (1.25e5 + 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-301. + 88.4i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (302. - 194. i)T + (9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (375. + 433. i)T + (-4.28e4 + 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-338. - 390. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (161. - 1.12e3i)T + (-3.73e5 - 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-8.88 + 2.60i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (166. + 363. i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-846. - 543. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-127. + 279. i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
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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
−12.96564392216280110480070990314, −11.68980667950818376702902995366, −10.98166100497338961504190695555, −9.199817559130533925230001047356, −8.467991376689632125025977563581, −6.96500128029763644332490527663, −6.00752765970930986179502321734, −5.01949572574208157905356533628, −3.46178407783862412697505978965, −1.03595937078740154528448644226,
1.63901937928396560798033101041, 3.49788711163746420518103127814, 4.68210031613268498054978536263, 6.13802935026487976093207507277, 6.97784994941009089204363197072, 8.912399695493394582119465988818, 9.989585307643181239647490605116, 10.75842538698569518097751441587, 11.78269147436223277052509304143, 12.55444923688023387580313192704
