| L(s) = 1 | + (0.149 − 0.988i)2-s + (−0.133 + 1.72i)3-s + (−0.955 − 0.294i)4-s + (0.273 + 3.65i)5-s + (1.68 + 0.388i)6-s + (−1.75 − 1.97i)7-s + (−0.433 + 0.900i)8-s + (−2.96 − 0.459i)9-s + (3.65 + 0.273i)10-s + (−4.00 + 1.57i)11-s + (0.636 − 1.61i)12-s + (3.06 + 2.44i)13-s + (−2.21 + 1.44i)14-s + (−6.34 − 0.0133i)15-s + (0.826 + 0.563i)16-s + (−0.681 − 0.632i)17-s + ⋯ |
| L(s) = 1 | + (0.105 − 0.699i)2-s + (−0.0768 + 0.997i)3-s + (−0.477 − 0.147i)4-s + (0.122 + 1.63i)5-s + (0.689 + 0.158i)6-s + (−0.664 − 0.747i)7-s + (−0.153 + 0.318i)8-s + (−0.988 − 0.153i)9-s + (1.15 + 0.0865i)10-s + (−1.20 + 0.473i)11-s + (0.183 − 0.465i)12-s + (0.850 + 0.678i)13-s + (−0.592 + 0.385i)14-s + (−1.63 − 0.00344i)15-s + (0.206 + 0.140i)16-s + (−0.165 − 0.153i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.199 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.581280 + 0.711272i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.581280 + 0.711272i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.149 + 0.988i)T \) |
| 3 | \( 1 + (0.133 - 1.72i)T \) |
| 7 | \( 1 + (1.75 + 1.97i)T \) |
| good | 5 | \( 1 + (-0.273 - 3.65i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (4.00 - 1.57i)T + (8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-3.06 - 2.44i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (0.681 + 0.632i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (3.09 - 1.78i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.13 - 2.29i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-8.21 - 1.87i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-6.51 - 3.76i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.70 - 1.14i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-4.98 - 2.40i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-7.93 + 3.82i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (6.41 + 0.966i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (-1.35 + 4.39i)T + (-43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.717 + 9.56i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.314 - 1.02i)T + (-50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (3.54 - 6.14i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.23 - 1.65i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-2.51 - 16.6i)T + (-69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (6.63 + 11.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.95 - 3.70i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-3.06 + 7.80i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 - 1.43iT - 97T^{2} \) |
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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
−11.66017005203790673155996386229, −10.76763000630378609807220404009, −10.38685961376956546421547860356, −9.766731376889777868593008396308, −8.441920481652308920799484979804, −7.01346552347467331797659835404, −6.09196548953205895217222448630, −4.59151882199874584651391340350, −3.48714846786497712249461957403, −2.66191220336375740130761433924,
0.65959671731143878536294484099, 2.72857619690796098813555194607, 4.71398852800493150886790245274, 5.73289549419039260369648739657, 6.30443850137981044848208044840, 7.88474134605464752788694265090, 8.485841713792759455681911597896, 9.081561893946445033904993368657, 10.56810415457720411647401845077, 11.98608594918333226971946442336
