| L(s) = 1 | + (−1.88 − 3.13i)2-s + (−1.65 + 2.50i)3-s + (−4.37 + 8.25i)4-s + (0.175 + 1.61i)5-s + (10.9 + 0.472i)6-s + (0.807 − 0.271i)7-s + (19.5 − 1.05i)8-s + (−3.51 − 8.28i)9-s + (4.72 − 3.59i)10-s + (−0.113 − 0.0769i)11-s + (−13.4 − 24.6i)12-s + (−4.49 − 4.25i)13-s + (−2.37 − 2.01i)14-s + (−4.33 − 2.23i)15-s + (−19.0 − 28.1i)16-s + (2.77 − 8.23i)17-s + ⋯ |
| L(s) = 1 | + (−0.941 − 1.56i)2-s + (−0.552 + 0.833i)3-s + (−1.09 + 2.06i)4-s + (0.0351 + 0.323i)5-s + (1.82 + 0.0787i)6-s + (0.115 − 0.0388i)7-s + (2.43 − 0.132i)8-s + (−0.390 − 0.920i)9-s + (0.472 − 0.359i)10-s + (−0.0103 − 0.00699i)11-s + (−1.11 − 2.05i)12-s + (−0.345 − 0.327i)13-s + (−0.169 − 0.143i)14-s + (−0.288 − 0.149i)15-s + (−1.19 − 1.75i)16-s + (0.163 − 0.484i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.893 + 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0993288 - 0.417883i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0993288 - 0.417883i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.65 - 2.50i)T \) |
| 59 | \( 1 + (11.2 - 57.9i)T \) |
| good | 2 | \( 1 + (1.88 + 3.13i)T + (-1.87 + 3.53i)T^{2} \) |
| 5 | \( 1 + (-0.175 - 1.61i)T + (-24.4 + 5.37i)T^{2} \) |
| 7 | \( 1 + (-0.807 + 0.271i)T + (39.0 - 29.6i)T^{2} \) |
| 11 | \( 1 + (0.113 + 0.0769i)T + (44.7 + 112. i)T^{2} \) |
| 13 | \( 1 + (4.49 + 4.25i)T + (9.14 + 168. i)T^{2} \) |
| 17 | \( 1 + (-2.77 + 8.23i)T + (-230. - 174. i)T^{2} \) |
| 19 | \( 1 + (5.30 + 32.3i)T + (-342. + 115. i)T^{2} \) |
| 23 | \( 1 + (21.1 - 5.86i)T + (453. - 272. i)T^{2} \) |
| 29 | \( 1 + (-3.98 + 6.61i)T + (-393. - 743. i)T^{2} \) |
| 31 | \( 1 + (3.27 - 19.9i)T + (-910. - 306. i)T^{2} \) |
| 37 | \( 1 + (-3.29 + 60.7i)T + (-1.36e3 - 148. i)T^{2} \) |
| 41 | \( 1 + (3.82 + 1.06i)T + (1.44e3 + 866. i)T^{2} \) |
| 43 | \( 1 + (32.6 + 48.1i)T + (-684. + 1.71e3i)T^{2} \) |
| 47 | \( 1 + (-9.16 + 84.2i)T + (-2.15e3 - 474. i)T^{2} \) |
| 53 | \( 1 + (-32.1 + 42.2i)T + (-751. - 2.70e3i)T^{2} \) |
| 61 | \( 1 + (-20.4 + 12.3i)T + (1.74e3 - 3.28e3i)T^{2} \) |
| 67 | \( 1 + (-4.15 - 76.6i)T + (-4.46e3 + 485. i)T^{2} \) |
| 71 | \( 1 + (-3.76 + 34.5i)T + (-4.92e3 - 1.08e3i)T^{2} \) |
| 73 | \( 1 + (-46.3 + 54.5i)T + (-862. - 5.25e3i)T^{2} \) |
| 79 | \( 1 + (7.03 - 17.6i)T + (-4.53e3 - 4.29e3i)T^{2} \) |
| 83 | \( 1 + (-65.3 - 141. i)T + (-4.45e3 + 5.25e3i)T^{2} \) |
| 89 | \( 1 + (-13.0 + 21.7i)T + (-3.71e3 - 6.99e3i)T^{2} \) |
| 97 | \( 1 + (71.8 + 84.5i)T + (-1.52e3 + 9.28e3i)T^{2} \) |
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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
−11.69071059221629024159141341869, −10.89010047080327342533365878108, −10.25769750483722546905255288624, −9.364588998862791908791056718148, −8.533971650241935238569242114450, −7.01920440509732338855439994551, −5.09941858038298456810169462516, −3.80409890443787057195280654162, −2.56145557678535665102726671301, −0.40334239320951353613383187081,
1.43050922430536042399955013204, 4.79396387771140470397399609911, 5.94056578337495174557751052564, 6.61036672254540856350896515668, 7.86704341146618920238248060752, 8.285812494290726637552998955597, 9.631862569906908898535267798412, 10.58175551403239862802446297783, 11.97183056030838094050297768645, 13.00355878250716656828664866628
