| L(s) = 1 | + (2.27 + 0.183i)2-s + (−0.845 − 0.534i)3-s + (3.14 + 0.511i)4-s + (2.05 + 1.82i)5-s + (−1.82 − 1.36i)6-s + (0.613 − 0.638i)7-s + (2.63 + 0.648i)8-s + (0.428 + 0.903i)9-s + (4.33 + 4.51i)10-s + (−2.07 + 4.37i)11-s + (−2.38 − 2.11i)12-s + (2.41 − 2.67i)13-s + (1.50 − 1.33i)14-s + (−0.764 − 2.64i)15-s + (−0.196 − 0.0657i)16-s + (−0.313 + 0.326i)17-s + ⋯ |
| L(s) = 1 | + (1.60 + 0.129i)2-s + (−0.487 − 0.308i)3-s + (1.57 + 0.255i)4-s + (0.920 + 0.815i)5-s + (−0.743 − 0.558i)6-s + (0.231 − 0.241i)7-s + (0.929 + 0.229i)8-s + (0.142 + 0.301i)9-s + (1.37 + 1.42i)10-s + (−0.625 + 1.31i)11-s + (−0.689 − 0.610i)12-s + (0.669 − 0.742i)13-s + (0.403 − 0.357i)14-s + (−0.197 − 0.681i)15-s + (−0.0492 − 0.0164i)16-s + (−0.0759 + 0.0791i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.25439 + 0.550841i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.25439 + 0.550841i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.845 + 0.534i)T \) |
| 13 | \( 1 + (-2.41 + 2.67i)T \) |
| good | 2 | \( 1 + (-2.27 - 0.183i)T + (1.97 + 0.320i)T^{2} \) |
| 5 | \( 1 + (-2.05 - 1.82i)T + (0.602 + 4.96i)T^{2} \) |
| 7 | \( 1 + (-0.613 + 0.638i)T + (-0.281 - 6.99i)T^{2} \) |
| 11 | \( 1 + (2.07 - 4.37i)T + (-6.95 - 8.52i)T^{2} \) |
| 17 | \( 1 + (0.313 - 0.326i)T + (-0.684 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.37 + 4.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.832 + 1.44i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.00 + 0.323i)T + (28.6 + 4.65i)T^{2} \) |
| 31 | \( 1 + (0.237 - 1.95i)T + (-30.0 - 7.41i)T^{2} \) |
| 37 | \( 1 + (-6.81 + 2.90i)T + (25.6 - 26.6i)T^{2} \) |
| 41 | \( 1 + (4.88 + 3.08i)T + (17.5 + 37.0i)T^{2} \) |
| 43 | \( 1 + (5.14 + 2.19i)T + (29.7 + 31.0i)T^{2} \) |
| 47 | \( 1 + (1.89 - 4.99i)T + (-35.1 - 31.1i)T^{2} \) |
| 53 | \( 1 + (-3.38 - 0.833i)T + (46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (8.53 - 2.84i)T + (47.1 - 35.4i)T^{2} \) |
| 61 | \( 1 + (-1.48 + 5.11i)T + (-51.5 - 32.6i)T^{2} \) |
| 67 | \( 1 + (4.59 - 0.747i)T + (63.5 - 21.2i)T^{2} \) |
| 71 | \( 1 + (-0.602 + 14.9i)T + (-70.7 - 5.71i)T^{2} \) |
| 73 | \( 1 + (-2.97 - 4.30i)T + (-25.8 + 68.2i)T^{2} \) |
| 79 | \( 1 + (-1.75 + 4.62i)T + (-59.1 - 52.3i)T^{2} \) |
| 83 | \( 1 + (10.1 + 5.30i)T + (47.1 + 68.3i)T^{2} \) |
| 89 | \( 1 + (-7.09 - 12.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.170 - 0.835i)T + (-89.2 + 38.0i)T^{2} \) |
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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
−11.04089479583471657063142863148, −10.50029695055454644210150138878, −9.442457814992138309930185494079, −7.70556782782077519239307582338, −6.92416147715776881277371907421, −6.13738523652904176570507364506, −5.33587558839325177800931584465, −4.48854272318616712998702384697, −3.07426200892920415715216828184, −2.05767574105647601573189036508,
1.67293505615390714690427516884, 3.21201611240360746383665545823, 4.27662219656307606754195864759, 5.44364907958991231698205011770, 5.63670443498293626156888539176, 6.57219878956486545000531602381, 8.208917751815060356984305683778, 9.182537716902619837748645204544, 10.20038776359938489720379370386, 11.38723719338282655051417339094
