| L(s) = 1 | + (0.929 − 1.06i)2-s + (−0.255 + 2.59i)3-s + (−0.270 − 1.98i)4-s + (−0.722 + 0.386i)5-s + (2.52 + 2.68i)6-s + (−0.195 − 0.980i)7-s + (−2.36 − 1.55i)8-s + (−3.72 − 0.741i)9-s + (−0.260 + 1.12i)10-s + (−3.09 + 3.77i)11-s + (5.21 − 0.195i)12-s + (0.980 − 1.83i)13-s + (−1.22 − 0.704i)14-s + (−0.817 − 1.97i)15-s + (−3.85 + 1.07i)16-s + (−2.99 + 7.22i)17-s + ⋯ |
| L(s) = 1 | + (0.657 − 0.753i)2-s + (−0.147 + 1.49i)3-s + (−0.135 − 0.990i)4-s + (−0.323 + 0.172i)5-s + (1.03 + 1.09i)6-s + (−0.0737 − 0.370i)7-s + (−0.835 − 0.549i)8-s + (−1.24 − 0.247i)9-s + (−0.0823 + 0.357i)10-s + (−0.933 + 1.13i)11-s + (1.50 − 0.0564i)12-s + (0.271 − 0.508i)13-s + (−0.327 − 0.188i)14-s + (−0.211 − 0.509i)15-s + (−0.963 + 0.268i)16-s + (−0.725 + 1.75i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.423230 + 0.824413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.423230 + 0.824413i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.929 + 1.06i)T \) |
| 7 | \( 1 + (0.195 + 0.980i)T \) |
| good | 3 | \( 1 + (0.255 - 2.59i)T + (-2.94 - 0.585i)T^{2} \) |
| 5 | \( 1 + (0.722 - 0.386i)T + (2.77 - 4.15i)T^{2} \) |
| 11 | \( 1 + (3.09 - 3.77i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.980 + 1.83i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (2.99 - 7.22i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.449 - 1.48i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-1.54 + 1.03i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (5.24 - 4.30i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-5.08 + 5.08i)T - 31iT^{2} \) |
| 37 | \( 1 + (6.43 + 1.95i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (5.43 + 8.13i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.567 - 5.76i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (5.47 + 2.26i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (6.20 + 5.09i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (-4.89 - 9.16i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-11.0 - 1.08i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-15.5 - 1.53i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (6.32 - 1.25i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.707 + 3.55i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-5.48 + 2.27i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-14.6 + 4.43i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (1.29 + 0.865i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (-0.303 + 0.303i)T - 97iT^{2} \) |
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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
−10.47668120675296419195236019417, −10.00051303484744785165116957852, −9.125065999747998491231394923383, −8.054622426859051120689637975788, −6.77391221946674397122766091712, −5.57326477275870842129068437662, −4.92444129137776465963496132854, −3.95674134045617486339425021128, −3.50518731365818627427823724212, −2.03315757086934411560734918600,
0.33853438489566109702788674762, 2.31918620394519571930331167532, 3.25696886464730418495339336728, 4.79618359736447372626226686915, 5.58170918758598416164139509383, 6.57675004444526139632897479273, 7.01328954197918282491515764008, 8.097007182974862094430045209233, 8.392929698715026839146863198948, 9.509636129909620900104290691480
