| L(s) = 1 | + (2.11 + 1.36i)3-s + (0.415 − 0.909i)5-s + (0.509 + 0.149i)7-s + (1.38 + 3.03i)9-s + (−3.18 + 3.67i)11-s + (5.61 − 1.64i)13-s + (2.11 − 1.36i)15-s + (0.910 + 6.33i)17-s + (0.565 − 3.92i)19-s + (0.874 + 1.00i)21-s + (−0.600 − 4.75i)23-s + (−0.654 − 0.755i)25-s + (−0.119 + 0.828i)27-s + (0.0305 + 0.212i)29-s + (−1.14 + 0.737i)31-s + ⋯ |
| L(s) = 1 | + (1.22 + 0.785i)3-s + (0.185 − 0.406i)5-s + (0.192 + 0.0564i)7-s + (0.461 + 1.01i)9-s + (−0.960 + 1.10i)11-s + (1.55 − 0.457i)13-s + (0.546 − 0.351i)15-s + (0.220 + 1.53i)17-s + (0.129 − 0.901i)19-s + (0.190 + 0.220i)21-s + (−0.125 − 0.992i)23-s + (−0.130 − 0.151i)25-s + (−0.0229 + 0.159i)27-s + (0.00566 + 0.0394i)29-s + (−0.206 + 0.132i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.01803 + 0.746393i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.01803 + 0.746393i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (0.600 + 4.75i)T \) |
| good | 3 | \( 1 + (-2.11 - 1.36i)T + (1.24 + 2.72i)T^{2} \) |
| 7 | \( 1 + (-0.509 - 0.149i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (3.18 - 3.67i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-5.61 + 1.64i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.910 - 6.33i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.565 + 3.92i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.0305 - 0.212i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (1.14 - 0.737i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.15 - 2.52i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-0.0105 + 0.0231i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (9.51 + 6.11i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 8.51T + 47T^{2} \) |
| 53 | \( 1 + (-7.70 - 2.26i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-6.75 + 1.98i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (6.51 - 4.18i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (7.29 + 8.42i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (2.48 + 2.86i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.02 + 7.12i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-3.25 + 0.955i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-2.96 - 6.49i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-13.1 - 8.47i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-0.237 + 0.520i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.70704718670387944431922820661, −10.27408581270479372119426478253, −9.275544754379502429622509429952, −8.399005141299575584186637527278, −8.035151427490701736256779856487, −6.52156905437374731984039568907, −5.20099343397808754638677107670, −4.23048130522121034939145568768, −3.21288463589518534439046600267, −1.91128066135253502188027772243,
1.48035435483145634433854909926, 2.86111855752749014837676006041, 3.59138968336859127381299493964, 5.40213209598127864797238297257, 6.44538936873638400503837813377, 7.54416831575875995215974610985, 8.156693798630055023193209165937, 8.937107546365865181040096005417, 9.915161589374920142132303127076, 11.08441496137957107944733182392
