| L(s) = 1 | + (−0.733 + 0.471i)2-s + (−0.142 + 0.989i)3-s + (−0.515 + 1.12i)4-s + (−1.20 + 0.353i)5-s + (−0.362 − 0.793i)6-s + (1.95 + 2.25i)7-s + (−0.402 − 2.79i)8-s + (−0.959 − 0.281i)9-s + (0.716 − 0.827i)10-s + (1.82 + 1.17i)11-s + (−1.04 − 0.670i)12-s + (2.53 − 2.92i)13-s + (−2.49 − 0.732i)14-s + (−0.178 − 1.24i)15-s + (−0.0108 − 0.0124i)16-s + (0.279 + 0.612i)17-s + ⋯ |
| L(s) = 1 | + (−0.518 + 0.333i)2-s + (−0.0821 + 0.571i)3-s + (−0.257 + 0.563i)4-s + (−0.538 + 0.158i)5-s + (−0.147 − 0.323i)6-s + (0.738 + 0.851i)7-s + (−0.142 − 0.988i)8-s + (−0.319 − 0.0939i)9-s + (0.226 − 0.261i)10-s + (0.551 + 0.354i)11-s + (−0.301 − 0.193i)12-s + (0.703 − 0.812i)13-s + (−0.666 − 0.195i)14-s + (−0.0461 − 0.320i)15-s + (−0.00270 − 0.00312i)16-s + (0.0678 + 0.148i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0750 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0750 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.444897 + 0.479649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.444897 + 0.479649i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (-0.811 + 4.72i)T \) |
| good | 2 | \( 1 + (0.733 - 0.471i)T + (0.830 - 1.81i)T^{2} \) |
| 5 | \( 1 + (1.20 - 0.353i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-1.95 - 2.25i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.82 - 1.17i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.53 + 2.92i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.279 - 0.612i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-1.71 + 3.74i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.63 - 7.96i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.717 - 4.98i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (8.56 + 2.51i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (5.37 - 1.57i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.20 + 8.41i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 8.72T + 47T^{2} \) |
| 53 | \( 1 + (7.25 + 8.36i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.81 + 4.39i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.635 + 4.42i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.74 + 3.04i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (6.66 - 4.28i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-2.65 + 5.80i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (4.70 - 5.43i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (11.4 + 3.35i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.0515 - 0.358i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (11.7 - 3.45i)T + (81.6 - 52.4i)T^{2} \) |
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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
−15.42715164149793275597999188964, −14.20577307233836507025168533315, −12.63169449677063909281448466568, −11.72416755152657748045143900956, −10.49003867641202909153942478446, −8.959280477397150275941044129367, −8.369357797564323128336025593427, −6.94263985955550845334257207861, −5.06327229462794589911848631116, −3.47652693971208393461936793251,
1.35185513953638859054842203424, 4.22989671920389193135822620658, 5.98613880124227611124814670046, 7.62312144244847558012473151778, 8.632974423979502788438074376193, 9.981639200515999395061888360608, 11.27888656566457935935461888977, 11.81813550931259242571323911513, 13.74470949649238115200505835310, 14.07832430624354326262547648475
