L(s) = 1 | + (2.35 + 0.537i)2-s + (−1.04 − 2.17i)3-s + (3.44 + 1.66i)4-s + (−0.222 + 0.974i)5-s + (−1.29 − 5.68i)6-s + (2.54 − 1.22i)7-s + (3.45 + 2.75i)8-s + (−1.76 + 2.21i)9-s + (−1.04 + 2.17i)10-s + (0.323 − 0.258i)11-s − 9.24i·12-s + (2.38 + 2.99i)13-s + (6.65 − 1.51i)14-s + (2.35 − 0.537i)15-s + (1.87 + 2.34i)16-s + 0.828i·17-s + ⋯ |
L(s) = 1 | + (1.66 + 0.379i)2-s + (−0.604 − 1.25i)3-s + (1.72 + 0.830i)4-s + (−0.0995 + 0.436i)5-s + (−0.529 − 2.31i)6-s + (0.963 − 0.463i)7-s + (1.22 + 0.973i)8-s + (−0.587 + 0.737i)9-s + (−0.331 + 0.687i)10-s + (0.0976 − 0.0778i)11-s − 2.66i·12-s + (0.662 + 0.830i)13-s + (1.77 − 0.406i)14-s + (0.607 − 0.138i)15-s + (0.467 + 0.586i)16-s + 0.200i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.51034 - 0.944457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.51034 - 0.944457i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
good | 2 | \( 1 + (-2.35 - 0.537i)T + (1.80 + 0.867i)T^{2} \) |
| 3 | \( 1 + (1.04 + 2.17i)T + (-1.87 + 2.34i)T^{2} \) |
| 5 | \( 1 + (0.222 - 0.974i)T + (-4.50 - 2.16i)T^{2} \) |
| 7 | \( 1 + (-2.54 + 1.22i)T + (4.36 - 5.47i)T^{2} \) |
| 11 | \( 1 + (-0.323 + 0.258i)T + (2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.38 - 2.99i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 - 0.828iT - 17T^{2} \) |
| 19 | \( 1 + (-2.60 + 5.40i)T + (-11.8 - 14.8i)T^{2} \) |
| 23 | \( 1 + (0.813 + 3.56i)T + (-20.7 + 9.97i)T^{2} \) |
| 31 | \( 1 + (9.81 + 2.24i)T + (27.9 + 13.4i)T^{2} \) |
| 37 | \( 1 + (-3.12 - 2.49i)T + (8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 - 4.48iT - 41T^{2} \) |
| 43 | \( 1 + (-3.49 + 0.797i)T + (38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (2.53 - 2.02i)T + (10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (2.11 - 9.24i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + 3.65T + 59T^{2} \) |
| 61 | \( 1 + (-2.09 - 4.35i)T + (-38.0 + 47.6i)T^{2} \) |
| 67 | \( 1 + (3.52 - 4.42i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (-5.50 - 6.90i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (3.89 - 0.890i)T + (65.7 - 31.6i)T^{2} \) |
| 79 | \( 1 + (1.88 + 1.50i)T + (17.5 + 77.0i)T^{2} \) |
| 83 | \( 1 + (6.89 + 3.32i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-12.1 - 2.77i)T + (80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 + (1.94 - 4.04i)T + (-60.4 - 75.8i)T^{2} \) |
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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
−10.88295190606124132385092666760, −9.089645111975705744755481299800, −7.78828854484648530138832501332, −7.17288753342412048660283866327, −6.58316981266034410041590783412, −5.82356020384711580818836880006, −4.84570300945459100825483647574, −3.99326496267774278669912434047, −2.67106685423758962235281285310, −1.38803769438630638393140045290,
1.74545550923616741903269325642, 3.36487921705165706217179104460, 4.00084343990645630425813355064, 5.06684916322179753409228883913, 5.32774917490225906050309145938, 6.09539225004731141024726138157, 7.60444789010429154126119615710, 8.698013460386253706084512128902, 9.753016925059137579642303658262, 10.75799980384828327414775413477