| 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
−13.22971373655365214533450719296, −11.97388256495022885234530207162, −11.61631124785661270756138936281, −10.29787437089573671956181225450, −9.423895393209313804842525038853, −8.150754913319245763446129843884, −6.40655643747305746286893787710, −4.50466522741825984773697685515, −3.57312716047285884889582544714, −2.15578584535344603299560178573,
3.28178555272531201854561675266, 4.87178539010652911212605913500, 5.94711542312137343108364507561, 7.27677498475682137856312634008, 8.173110020674918630063211937924, 8.944073258600876713699982804154, 10.78380431425677358518920042387, 12.09837361266422842262594854195, 13.46781982267322184627205802786, 13.80967112472482572246420492123
