| L(s) = 1 | + (−1.12 + 0.856i)2-s + (0.434 + 1.67i)3-s + (0.531 − 1.92i)4-s + (−2.13 − 1.13i)5-s + (−1.92 − 1.51i)6-s + (1.86 + 0.370i)7-s + (1.05 + 2.62i)8-s + (−2.62 + 1.45i)9-s + (3.37 − 0.544i)10-s + (−1.11 + 0.915i)11-s + (3.46 + 0.0528i)12-s + (−3.82 + 2.04i)13-s + (−2.41 + 1.17i)14-s + (0.984 − 4.06i)15-s + (−3.43 − 2.04i)16-s + (−4.29 + 1.77i)17-s + ⋯ |
| L(s) = 1 | + (−0.795 + 0.605i)2-s + (0.250 + 0.968i)3-s + (0.265 − 0.964i)4-s + (−0.953 − 0.509i)5-s + (−0.786 − 0.618i)6-s + (0.703 + 0.139i)7-s + (0.372 + 0.927i)8-s + (−0.874 + 0.485i)9-s + (1.06 − 0.172i)10-s + (−0.336 + 0.276i)11-s + (0.999 + 0.0152i)12-s + (−1.05 + 0.566i)13-s + (−0.644 + 0.315i)14-s + (0.254 − 1.05i)15-s + (−0.858 − 0.512i)16-s + (−1.04 + 0.431i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0644214 - 0.371415i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0644214 - 0.371415i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.12 - 0.856i)T \) |
| 3 | \( 1 + (-0.434 - 1.67i)T \) |
| good | 5 | \( 1 + (2.13 + 1.13i)T + (2.77 + 4.15i)T^{2} \) |
| 7 | \( 1 + (-1.86 - 0.370i)T + (6.46 + 2.67i)T^{2} \) |
| 11 | \( 1 + (1.11 - 0.915i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (3.82 - 2.04i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (4.29 - 1.77i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (1.29 - 4.25i)T + (-15.7 - 10.5i)T^{2} \) |
| 23 | \( 1 + (5.93 + 3.96i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-7.57 - 6.21i)T + (5.65 + 28.4i)T^{2} \) |
| 31 | \( 1 + (-2.40 + 2.40i)T - 31iT^{2} \) |
| 37 | \( 1 + (2.34 + 7.73i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-3.34 - 2.23i)T + (15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (0.763 - 7.75i)T + (-42.1 - 8.38i)T^{2} \) |
| 47 | \( 1 + (-1.44 + 0.597i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (2.65 - 2.17i)T + (10.3 - 51.9i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.463i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-1.05 - 10.6i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (14.9 - 1.47i)T + (65.7 - 13.0i)T^{2} \) |
| 71 | \( 1 + (-6.43 - 1.27i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.288 - 1.45i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-2.98 + 7.21i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (1.82 - 6.01i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-5.69 - 8.52i)T + (-34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-3.88 - 3.88i)T + 97iT^{2} \) |
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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
−11.60837330850383034178067181516, −10.66054092551929493605675995508, −9.968817886293854051812384686250, −8.868572162445969161077796280149, −8.282534662735793991576154522740, −7.57534348409178191986581676624, −6.15949373489458850860492662612, −4.79465583206292887621037482913, −4.32960476867783254812556696160, −2.23632234247003637203674996080,
0.29524227622337099904667412133, 2.18976499378829913896584607172, 3.17810088627299410489372884031, 4.64222637298755249178935390028, 6.53120964259699945822911604084, 7.46760701348941484936308477254, 7.953063738309687940120991072633, 8.755511100090569852582166985252, 10.00914241978787791122957601284, 11.04543683932358161984497474802
