L(s) = 1 | + (2.17 − 2.06i)3-s + (3.17 − 3.17i)5-s − 6.03i·7-s + (0.485 − 8.98i)9-s + (−13.0 + 13.0i)11-s + (6.39 − 6.39i)13-s + (0.363 − 13.4i)15-s + 4.39i·17-s + (3.21 − 3.21i)19-s + (−12.4 − 13.1i)21-s + 34.0·23-s + 4.78i·25-s + (−17.4 − 20.5i)27-s + (−27.9 − 27.9i)29-s + 7.90·31-s + ⋯ |
L(s) = 1 | + (0.725 − 0.687i)3-s + (0.635 − 0.635i)5-s − 0.862i·7-s + (0.0539 − 0.998i)9-s + (−1.18 + 1.18i)11-s + (0.491 − 0.491i)13-s + (0.0242 − 0.898i)15-s + 0.258i·17-s + (0.168 − 0.168i)19-s + (−0.593 − 0.626i)21-s + 1.47·23-s + 0.191i·25-s + (−0.647 − 0.761i)27-s + (−0.964 − 0.964i)29-s + 0.255·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.53112 - 1.26415i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53112 - 1.26415i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.17 + 2.06i)T \) |
good | 5 | \( 1 + (-3.17 + 3.17i)T - 25iT^{2} \) |
| 7 | \( 1 + 6.03iT - 49T^{2} \) |
| 11 | \( 1 + (13.0 - 13.0i)T - 121iT^{2} \) |
| 13 | \( 1 + (-6.39 + 6.39i)T - 169iT^{2} \) |
| 17 | \( 1 - 4.39iT - 289T^{2} \) |
| 19 | \( 1 + (-3.21 + 3.21i)T - 361iT^{2} \) |
| 23 | \( 1 - 34.0T + 529T^{2} \) |
| 29 | \( 1 + (27.9 + 27.9i)T + 841iT^{2} \) |
| 31 | \( 1 - 7.90T + 961T^{2} \) |
| 37 | \( 1 + (-20.0 - 20.0i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 45.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36.0 - 36.0i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 5.08iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (20.7 - 20.7i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (39.0 - 39.0i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (49.8 - 49.8i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (44.9 - 44.9i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 46.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 97.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 40.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + (35.5 + 35.5i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 69.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 61.0T + 9.40e3T^{2} \) |
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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
−12.59426330109254621705322051262, −11.03842137280409114031270013727, −9.930733010821151167064635009855, −9.109132859957808072916018223677, −7.86360105030038803463351046466, −7.21473460148531006120139932740, −5.77470929172252338410038674655, −4.40065953840858519181279413818, −2.71600721040980988087654493237, −1.19614818974988754487553824973,
2.39815259102770747512462884065, 3.32117336974028596953147300227, 5.10156430044307350049204845935, 6.05105756428710128706788150325, 7.59532738708781124918029221649, 8.755938121045881840806590064760, 9.380849025806994772674860264757, 10.67811866638664078663516971559, 11.10109720522352243086891807525, 12.74618192683454900347124548626