| L(s) = 1 | + i·3-s + 7i·7-s + 26·9-s − 7·11-s − 23i·13-s + 25i·17-s + 62·19-s − 7·21-s − 86i·23-s + 53i·27-s + 29·29-s − 12·31-s − 7i·33-s + 150i·37-s + 23·39-s + ⋯ |
| L(s) = 1 | + 0.192i·3-s + 0.377i·7-s + 0.962·9-s − 0.191·11-s − 0.490i·13-s + 0.356i·17-s + 0.748·19-s − 0.0727·21-s − 0.779i·23-s + 0.377i·27-s + 0.185·29-s − 0.0695·31-s − 0.0369i·33-s + 0.666i·37-s + 0.0944·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.186829750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.186829750\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
| good | 3 | \( 1 - iT - 27T^{2} \) |
| 11 | \( 1 + 7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 25iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 62T + 6.85e3T^{2} \) |
| 23 | \( 1 + 86iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 29T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12T + 2.97e4T^{2} \) |
| 37 | \( 1 - 150iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 204T + 6.89e4T^{2} \) |
| 43 | \( 1 + 178iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 33iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 452iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 120T + 2.05e5T^{2} \) |
| 61 | \( 1 - 920T + 2.26e5T^{2} \) |
| 67 | \( 1 - 300iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 520T + 3.57e5T^{2} \) |
| 73 | \( 1 - 370iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 636iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 292T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3iT - 9.12e5T^{2} \) |
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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
−10.13580241389257044186361624708, −9.349417299713022560125209002153, −8.386369631358247173761107018320, −7.52899976365169017191987097715, −6.60443875788375561258255530790, −5.55452153637570970004715625874, −4.63821095685446429097621657659, −3.57705631509768611050649491747, −2.35304843500013363358561034160, −0.953898241619002643545702983230,
0.819270138761063546520282987797, 2.01503445543806695992159059438, 3.44841923958752271731378756645, 4.43891753240379548869642583566, 5.43052101960937810054674137538, 6.63990951691046760910314853794, 7.33041749886971644816062983380, 8.089207104417518486162025165945, 9.355947215599249852431263773139, 9.846969556068472240794016135431
