| L(s) = 1 | + (3.16 + 1.44i)2-s + (−1.48 + 5.05i)3-s + (5.28 + 6.10i)4-s + (3.33 − 3.72i)5-s + (−11.9 + 13.8i)6-s + (1.58 − 11.0i)7-s + (3.98 + 13.5i)8-s + (−15.7 − 10.1i)9-s + (15.9 − 6.93i)10-s + (−6.06 + 2.77i)11-s + (−38.6 + 17.6i)12-s + (0.446 − 0.0642i)13-s + (20.9 − 32.5i)14-s + (13.8 + 22.3i)15-s + (−2.40 + 16.7i)16-s + (4.82 − 5.57i)17-s + ⋯ |
| L(s) = 1 | + (1.58 + 0.721i)2-s + (−0.494 + 1.68i)3-s + (1.32 + 1.52i)4-s + (0.667 − 0.744i)5-s + (−1.99 + 2.30i)6-s + (0.226 − 1.57i)7-s + (0.498 + 1.69i)8-s + (−1.74 − 1.12i)9-s + (1.59 − 0.693i)10-s + (−0.551 + 0.251i)11-s + (−3.22 + 1.47i)12-s + (0.0343 − 0.00494i)13-s + (1.49 − 2.32i)14-s + (0.922 + 1.49i)15-s + (−0.150 + 1.04i)16-s + (0.284 − 0.327i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.75593 + 2.19525i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.75593 + 2.19525i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-3.33 + 3.72i)T \) |
| 23 | \( 1 + (-2.51 - 22.8i)T \) |
| good | 2 | \( 1 + (-3.16 - 1.44i)T + (2.61 + 3.02i)T^{2} \) |
| 3 | \( 1 + (1.48 - 5.05i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (-1.58 + 11.0i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (6.06 - 2.77i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (-0.446 + 0.0642i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (-4.82 + 5.57i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (18.7 - 16.2i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (-3.14 + 3.62i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (-0.998 + 0.293i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (-23.1 - 14.8i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (58.3 - 37.4i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (-2.13 - 0.626i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 - 47.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-11.6 + 81.1i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (-6.20 - 43.1i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (2.80 + 9.53i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-23.1 + 50.7i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (-17.3 + 37.8i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (-34.2 + 29.6i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (8.92 - 1.28i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-64.4 - 41.4i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-24.7 + 84.3i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (140. - 90.3i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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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
−13.78417405424954080862592954520, −12.95601737956824894772570262912, −11.66013054869282547817992327681, −10.51682047021950632568261531048, −9.716189860460381129381841898537, −7.953659136469424657989877800038, −6.35695868994748008460750868569, −5.20403330958049686349298591824, −4.54899263712120227992781570501, −3.61519767238855491607161065463,
2.05302162894541414454459162966, 2.69539714665715011505883950770, 5.28220056114580865576928169528, 6.00261368962309793462448549756, 6.80933323040773148029137529241, 8.530335081722432532152582964107, 10.61311523793158134976796907811, 11.44002136304581520922083395116, 12.30332394986754403876585054044, 12.90487835317325262871131900122
