L(s) = 1 | + (1.30 − 0.554i)2-s + (1.38 − 1.44i)4-s + (0.345 − 0.597i)5-s + (2.63 − 0.222i)7-s + (1.00 − 2.64i)8-s + (0.117 − 0.969i)10-s + (1.63 + 2.82i)11-s − 5.27·13-s + (3.30 − 1.75i)14-s + (−0.167 − 3.99i)16-s + (2.20 − 1.27i)17-s + (−0.484 − 0.279i)19-s + (−0.385 − 1.32i)20-s + (3.68 + 2.76i)22-s + (−2.50 − 1.44i)23-s + ⋯ |
L(s) = 1 | + (0.919 − 0.392i)2-s + (0.692 − 0.721i)4-s + (0.154 − 0.267i)5-s + (0.996 − 0.0840i)7-s + (0.353 − 0.935i)8-s + (0.0370 − 0.306i)10-s + (0.491 + 0.851i)11-s − 1.46·13-s + (0.883 − 0.468i)14-s + (−0.0417 − 0.999i)16-s + (0.534 − 0.308i)17-s + (−0.111 − 0.0642i)19-s + (−0.0861 − 0.296i)20-s + (0.786 + 0.590i)22-s + (−0.522 − 0.301i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.599 + 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.39846 - 1.19967i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.39846 - 1.19967i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.30 + 0.554i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.63 + 0.222i)T \) |
good | 5 | \( 1 + (-0.345 + 0.597i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.63 - 2.82i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 5.27T + 13T^{2} \) |
| 17 | \( 1 + (-2.20 + 1.27i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.484 + 0.279i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.50 + 1.44i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 0.444iT - 29T^{2} \) |
| 31 | \( 1 + (4.45 + 7.71i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.00 - 3.46i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 9.76iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (-2.20 + 3.81i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (8.17 - 4.71i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (8.59 - 4.96i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.23 - 9.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.45 + 2.51i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.29iT - 71T^{2} \) |
| 73 | \( 1 + (5.28 - 3.05i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.01 - 2.89i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.83iT - 83T^{2} \) |
| 89 | \( 1 + (1.5 + 0.866i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7.42iT - 97T^{2} \) |
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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
−11.02659431655391913594122037555, −9.953197599415778397132361059862, −9.356826917121554140782743789268, −7.79439445319447056593983863399, −7.14092658243805035443580486555, −5.85810912058314718269784127633, −4.85261581822579691484141070160, −4.29808354456778946397516444632, −2.67728737996027828370331919080, −1.52565051987157310996247378648,
1.99983118682972407606865484212, 3.27779596138057874068582937038, 4.50016354749647425896901898005, 5.37075351838768075426719740669, 6.29643372604631269766517368915, 7.37928097433444601372607338226, 8.068081137721308218909779164783, 9.115967990348891327233117386158, 10.47428306799914325275094847857, 11.17299063964792143127830779171