| L(s) = 1 | + (3.25 + 1.34i)2-s + (−0.337 − 1.69i)3-s + (5.95 + 5.95i)4-s + (−6.07 − 4.05i)5-s + (1.19 − 5.98i)6-s + (−6.13 + 4.10i)7-s + (5.95 + 14.3i)8-s + (−2.77 + 1.14i)9-s + (−14.2 − 21.3i)10-s + (14.1 + 2.81i)11-s + (8.09 − 12.1i)12-s + (−2.95 + 2.95i)13-s + (−25.5 + 5.07i)14-s + (−4.84 + 11.6i)15-s + 21.1i·16-s + (15.0 − 7.84i)17-s + ⋯ |
| L(s) = 1 | + (1.62 + 0.674i)2-s + (−0.112 − 0.566i)3-s + (1.48 + 1.48i)4-s + (−1.21 − 0.811i)5-s + (0.198 − 0.997i)6-s + (−0.877 + 0.586i)7-s + (0.744 + 1.79i)8-s + (−0.307 + 0.127i)9-s + (−1.42 − 2.13i)10-s + (1.28 + 0.255i)11-s + (0.674 − 1.01i)12-s + (−0.227 + 0.227i)13-s + (−1.82 + 0.362i)14-s + (−0.322 + 0.779i)15-s + 1.32i·16-s + (0.887 − 0.461i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.99764 + 0.428795i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.99764 + 0.428795i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.337 + 1.69i)T \) |
| 17 | \( 1 + (-15.0 + 7.84i)T \) |
| good | 2 | \( 1 + (-3.25 - 1.34i)T + (2.82 + 2.82i)T^{2} \) |
| 5 | \( 1 + (6.07 + 4.05i)T + (9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (6.13 - 4.10i)T + (18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (-14.1 - 2.81i)T + (111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (2.95 - 2.95i)T - 169iT^{2} \) |
| 19 | \( 1 + (1.86 + 0.772i)T + (255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-6.85 + 34.4i)T + (-488. - 202. i)T^{2} \) |
| 29 | \( 1 + (25.4 - 38.0i)T + (-321. - 776. i)T^{2} \) |
| 31 | \( 1 + (31.0 - 6.17i)T + (887. - 367. i)T^{2} \) |
| 37 | \( 1 + (-7.50 - 37.7i)T + (-1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (1.47 - 0.983i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (7.29 - 3.02i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-14.3 + 14.3i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (38.8 + 16.0i)T + (1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-4.57 - 11.0i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-25.3 - 37.9i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 + 45.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (20.2 + 101. i)T + (-4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (22.1 + 14.8i)T + (2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-41.0 - 8.17i)T + (5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (-13.5 + 32.7i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (26.7 + 26.7i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (8.82 - 13.2i)T + (-3.60e3 - 8.69e3i)T^{2} \) |
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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
−15.13023750570543578078083935023, −14.31881990285969574969271291776, −12.81890590524105239690768231516, −12.35172991523599078036609673932, −11.61888964864144621990738809288, −8.926169671769798744803007997507, −7.41584652838746840247104454605, −6.37461795833125223726566940383, −4.86247243829688446072567628762, −3.46551384170403980091035648809,
3.48143439997436382196780482665, 3.91402305685453219631735713353, 5.90199301265028465038425928916, 7.27405121031892006627032451834, 9.788444534111015629076893491165, 11.06851434947001475201713600787, 11.68324220755762491707815183864, 12.84519571368643644239931872193, 14.16232732416293618071697708702, 14.93590375257791630778031028847
