| L(s) = 1 | + (2.47 − 1.07i)2-s + (−0.351 + 5.13i)3-s + (−0.507 + 0.543i)4-s + (−4.27 + 2.59i)5-s + (4.64 + 13.0i)6-s + (−4.33 − 20.8i)7-s + (−7.88 + 22.1i)8-s + (0.466 + 0.0641i)9-s + (−7.76 + 11.0i)10-s + (−23.0 + 18.7i)11-s + (−2.61 − 2.79i)12-s + (−47.4 − 13.2i)13-s + (−33.1 − 46.8i)14-s + (−11.8 − 22.8i)15-s + (3.92 + 57.3i)16-s + (63.2 + 51.4i)17-s + ⋯ |
| L(s) = 1 | + (0.873 − 0.379i)2-s + (−0.0676 + 0.988i)3-s + (−0.0634 + 0.0679i)4-s + (−0.382 + 0.232i)5-s + (0.316 + 0.889i)6-s + (−0.234 − 1.12i)7-s + (−0.348 + 0.980i)8-s + (0.0172 + 0.00237i)9-s + (−0.245 + 0.347i)10-s + (−0.631 + 0.513i)11-s + (−0.0628 − 0.0673i)12-s + (−1.01 − 0.283i)13-s + (−0.631 − 0.895i)14-s + (−0.203 − 0.393i)15-s + (0.0613 + 0.896i)16-s + (0.902 + 0.734i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0343093 + 0.716013i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0343093 + 0.716013i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (4.27 - 2.59i)T \) |
| 47 | \( 1 + (-199. - 252. i)T \) |
| good | 2 | \( 1 + (-2.47 + 1.07i)T + (5.46 - 5.84i)T^{2} \) |
| 3 | \( 1 + (0.351 - 5.13i)T + (-26.7 - 3.67i)T^{2} \) |
| 7 | \( 1 + (4.33 + 20.8i)T + (-314. + 136. i)T^{2} \) |
| 11 | \( 1 + (23.0 - 18.7i)T + (270. - 1.30e3i)T^{2} \) |
| 13 | \( 1 + (47.4 + 13.2i)T + (1.87e3 + 1.14e3i)T^{2} \) |
| 17 | \( 1 + (-63.2 - 51.4i)T + (9.99e2 + 4.81e3i)T^{2} \) |
| 19 | \( 1 + (108. + 65.9i)T + (3.15e3 + 6.09e3i)T^{2} \) |
| 23 | \( 1 + (145. + 63.3i)T + (8.30e3 + 8.89e3i)T^{2} \) |
| 29 | \( 1 + (135. - 37.9i)T + (2.08e4 - 1.26e4i)T^{2} \) |
| 31 | \( 1 + (16.2 + 238. i)T + (-2.95e4 + 4.05e3i)T^{2} \) |
| 37 | \( 1 + (94.5 - 133. i)T + (-1.69e4 - 4.77e4i)T^{2} \) |
| 41 | \( 1 + (-139. - 391. i)T + (-5.34e4 + 4.34e4i)T^{2} \) |
| 43 | \( 1 + (-199. + 213. i)T + (-5.42e3 - 7.93e4i)T^{2} \) |
| 53 | \( 1 + (-208. - 587. i)T + (-1.15e5 + 9.39e4i)T^{2} \) |
| 59 | \( 1 + (-143. - 153. i)T + (-1.40e4 + 2.04e5i)T^{2} \) |
| 61 | \( 1 + (-228. - 324. i)T + (-7.60e4 + 2.13e5i)T^{2} \) |
| 67 | \( 1 + (27.7 - 133. i)T + (-2.75e5 - 1.19e5i)T^{2} \) |
| 71 | \( 1 + (-495. - 215. i)T + (2.44e5 + 2.61e5i)T^{2} \) |
| 73 | \( 1 + (109. - 15.1i)T + (3.74e5 - 1.04e5i)T^{2} \) |
| 79 | \( 1 + (-519. - 1.00e3i)T + (-2.84e5 + 4.02e5i)T^{2} \) |
| 83 | \( 1 + (884. - 719. i)T + (1.16e5 - 5.59e5i)T^{2} \) |
| 89 | \( 1 + (1.20e3 - 735. i)T + (3.24e5 - 6.25e5i)T^{2} \) |
| 97 | \( 1 + (-124. + 1.82e3i)T + (-9.04e5 - 1.24e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.43582777515697734132811286376, −11.11346813190229012341904731621, −10.40388224540028836694688240765, −9.684310114800580538561988590289, −8.161549044494633670966307920463, −7.24571875534284912190161874458, −5.60787980121221019611621891123, −4.22671056606219159696626485769, −4.15740228600254846995465641620, −2.61043271947054319483009280667,
0.20831682715615135805425063218, 2.15681090043070739415663683628, 3.78304271897797954475119464828, 5.23994125759434170261746095052, 5.92633052733288489373278635191, 7.05084515792223112719079215140, 8.005446306999471125401036254156, 9.209790074015520471176114989305, 10.25131510372764733403764610659, 11.95941359279861027159789013153
