| L(s) = 1 | + (−1.02 + 0.744i)2-s + (−1.35 + 0.603i)3-s + (−0.122 + 0.376i)4-s + (1.90 + 3.29i)5-s + (0.940 − 1.62i)6-s + (1.46 + 1.62i)7-s + (−0.937 − 2.88i)8-s + (−0.533 + 0.592i)9-s + (−4.39 − 1.95i)10-s + (−0.929 + 0.197i)11-s + (−0.0613 − 0.584i)12-s + (0.0175 − 0.167i)13-s + (−2.71 − 0.576i)14-s + (−4.56 − 3.31i)15-s + (2.46 + 1.79i)16-s + (−6.43 − 1.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.724 + 0.526i)2-s + (−0.782 + 0.348i)3-s + (−0.0611 + 0.188i)4-s + (0.849 + 1.47i)5-s + (0.383 − 0.664i)6-s + (0.553 + 0.614i)7-s + (−0.331 − 1.02i)8-s + (−0.177 + 0.197i)9-s + (−1.39 − 0.619i)10-s + (−0.280 + 0.0595i)11-s + (−0.0177 − 0.168i)12-s + (0.00487 − 0.0463i)13-s + (−0.724 − 0.154i)14-s + (−1.17 − 0.856i)15-s + (0.617 + 0.448i)16-s + (−1.56 − 0.331i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.284414 - 0.444056i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.284414 - 0.444056i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (1.02 - 0.744i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (1.35 - 0.603i)T + (2.00 - 2.22i)T^{2} \) |
| 5 | \( 1 + (-1.90 - 3.29i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.46 - 1.62i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (0.929 - 0.197i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.0175 + 0.167i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (6.43 + 1.36i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.120 - 1.14i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.43 - 4.40i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.08 + 0.785i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (1.93 - 3.35i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.299 - 0.133i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-1.00 - 9.57i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (4.56 + 3.31i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-4.90 + 5.44i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (2.42 - 1.07i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 - 1.74T + 61T^{2} \) |
| 67 | \( 1 + (0.276 + 0.478i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.760 + 0.844i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (7.74 - 1.64i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-4.44 - 0.944i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (0.298 + 0.132i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (-4.54 + 13.9i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.79 + 14.7i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.43393949036625728999525085153, −9.835772851114916669370412647135, −8.944974513432220833154137624411, −8.077167038503805113081613705551, −7.11984225336963655773309221855, −6.42799398314418695766419445065, −5.69943340511146620100669128405, −4.67358055381733514108467750347, −3.18409150264322877628392313302, −2.15348967293802311576031020395,
0.36510858813843074723167484179, 1.27753564718423573256502125859, 2.28633066734431371833677633384, 4.41685871617348188205767081671, 5.12676785352398802607071224732, 5.86616944578993897379910382975, 6.78251507775457179367180647015, 8.161112388640195900697741032180, 8.910688889736165981822451728675, 9.270668427917456273606031071621
