| L(s) = 1 | + (0.925 + 2.02i)2-s + (0.959 − 0.281i)3-s + (−1.94 + 2.24i)4-s + (−0.841 − 0.540i)5-s + (1.45 + 1.68i)6-s + (−0.0125 − 0.0872i)7-s + (−2.06 − 0.607i)8-s + (−1.68 + 1.08i)9-s + (0.317 − 2.20i)10-s + (1.01 − 2.21i)11-s + (−1.23 + 2.69i)12-s + (0.512 − 3.56i)13-s + (0.165 − 0.106i)14-s + (−0.959 − 0.281i)15-s + (0.160 + 1.11i)16-s + (−2.21 − 2.55i)17-s + ⋯ |
| L(s) = 1 | + (0.654 + 1.43i)2-s + (0.553 − 0.162i)3-s + (−0.971 + 1.12i)4-s + (−0.376 − 0.241i)5-s + (0.595 + 0.687i)6-s + (−0.00474 − 0.0329i)7-s + (−0.731 − 0.214i)8-s + (−0.560 + 0.360i)9-s + (0.100 − 0.697i)10-s + (0.305 − 0.669i)11-s + (−0.355 + 0.779i)12-s + (0.142 − 0.988i)13-s + (0.0441 − 0.0283i)14-s + (−0.247 − 0.0727i)15-s + (0.0402 + 0.279i)16-s + (−0.537 − 0.620i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0132 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0132 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04721 + 1.06122i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04721 + 1.06122i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (4.28 - 2.15i)T \) |
| good | 2 | \( 1 + (-0.925 - 2.02i)T + (-1.30 + 1.51i)T^{2} \) |
| 3 | \( 1 + (-0.959 + 0.281i)T + (2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (0.0125 + 0.0872i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.01 + 2.21i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.512 + 3.56i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (2.21 + 2.55i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-1.87 + 2.16i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-4.87 - 5.62i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.65 - 0.485i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-0.595 + 0.382i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (4.66 + 3.00i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (-2.30 + 0.675i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + (-1.47 - 10.2i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (1.64 - 11.4i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (3.89 + 1.14i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (5.07 + 11.1i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.787 - 1.72i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-3.61 + 4.16i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.997 - 6.93i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-3.51 + 2.25i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-15.3 + 4.49i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-12.0 - 7.77i)T + (40.2 + 88.2i)T^{2} \) |
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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
−13.80967112472482572246420492123, −13.46781982267322184627205802786, −12.09837361266422842262594854195, −10.78380431425677358518920042387, −8.944073258600876713699982804154, −8.173110020674918630063211937924, −7.27677498475682137856312634008, −5.94711542312137343108364507561, −4.87178539010652911212605913500, −3.28178555272531201854561675266,
2.15578584535344603299560178573, 3.57312716047285884889582544714, 4.50466522741825984773697685515, 6.40655643747305746286893787710, 8.150754913319245763446129843884, 9.423895393209313804842525038853, 10.29787437089573671956181225450, 11.61631124785661270756138936281, 11.97388256495022885234530207162, 13.22971373655365214533450719296
