L(s) = 1 | + (1.58 + 0.687i)3-s + (−0.300 + 0.520i)5-s + (0.5 + 0.866i)7-s + (2.05 + 2.18i)9-s + (0.800 + 1.38i)11-s + (−0.165 + 0.286i)13-s + (−0.834 + 0.620i)15-s − 1.44·17-s + 2.57·19-s + (0.199 + 1.72i)21-s + (−0.924 + 1.60i)23-s + (2.31 + 4.01i)25-s + (1.76 + 4.88i)27-s + (−1.75 − 3.04i)29-s + (4.81 − 8.33i)31-s + ⋯ |
L(s) = 1 | + (0.917 + 0.396i)3-s + (−0.134 + 0.232i)5-s + (0.188 + 0.327i)7-s + (0.685 + 0.728i)9-s + (0.241 + 0.417i)11-s + (−0.0458 + 0.0793i)13-s + (−0.215 + 0.160i)15-s − 0.351·17-s + 0.591·19-s + (0.0435 + 0.375i)21-s + (−0.192 + 0.333i)23-s + (0.463 + 0.803i)25-s + (0.339 + 0.940i)27-s + (−0.326 − 0.565i)29-s + (0.863 − 1.49i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.598 - 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.73486 + 0.869581i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73486 + 0.869581i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.58 - 0.687i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.300 - 0.520i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.800 - 1.38i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.165 - 0.286i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 + (0.924 - 1.60i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.75 + 3.04i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.81 + 8.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.600T + 37T^{2} \) |
| 41 | \( 1 + (3.31 - 5.73i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.81 + 3.14i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.95 + 3.38i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 9.27T + 53T^{2} \) |
| 59 | \( 1 + (-6.93 + 12.0i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.59 - 4.49i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.90 + 10.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 - 4.13T + 73T^{2} \) |
| 79 | \( 1 + (4.06 + 7.04i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.78 - 4.83i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3.83T + 89T^{2} \) |
| 97 | \( 1 + (-0.974 - 1.68i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.06065457777215122309986117411, −9.859003540712379400774942057749, −9.427276179538353006916112949517, −8.343299546177209055565712416268, −7.63144900985958841492917654532, −6.60932612144543570425636573096, −5.22770603655889598112775101207, −4.20408709121265937108293377693, −3.12389423794832267151848038583, −1.91813160399567189891598465733,
1.21493860260094065086161520857, 2.74277733912044740386131447705, 3.83656885352081999245845223984, 4.95495717128829265056161951895, 6.42175640834086552801502056504, 7.21800919589424456466248493210, 8.255224135087342022993990377975, 8.778640244435453988143116202373, 9.806081909362102679777744490480, 10.69597673350873703206910201563