| L(s) = 1 | + (−1.91 − 1.38i)2-s + (1.19 − 3.68i)3-s + (−0.748 − 2.30i)4-s + (12.0 − 8.75i)5-s + (−7.39 + 5.37i)6-s + (−2.16 − 6.65i)7-s + (−7.60 + 23.4i)8-s + (9.71 + 7.05i)9-s − 35.1·10-s + (−8.51 − 35.4i)11-s − 9.38·12-s + (−24.9 − 18.1i)13-s + (−5.10 + 15.7i)14-s + (−17.8 − 54.8i)15-s + (31.3 − 22.7i)16-s + (−79.6 + 57.8i)17-s + ⋯ |
| L(s) = 1 | + (−0.675 − 0.490i)2-s + (0.230 − 0.708i)3-s + (−0.0936 − 0.288i)4-s + (1.07 − 0.783i)5-s + (−0.503 + 0.365i)6-s + (−0.116 − 0.359i)7-s + (−0.336 + 1.03i)8-s + (0.359 + 0.261i)9-s − 1.11·10-s + (−0.233 − 0.972i)11-s − 0.225·12-s + (−0.532 − 0.386i)13-s + (−0.0975 + 0.300i)14-s + (−0.306 − 0.944i)15-s + (0.489 − 0.355i)16-s + (−1.13 + 0.825i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.542i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.839 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.327806 - 1.11145i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.327806 - 1.11145i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.16 + 6.65i)T \) |
| 11 | \( 1 + (8.51 + 35.4i)T \) |
| good | 2 | \( 1 + (1.91 + 1.38i)T + (2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (-1.19 + 3.68i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (-12.0 + 8.75i)T + (38.6 - 118. i)T^{2} \) |
| 13 | \( 1 + (24.9 + 18.1i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (79.6 - 57.8i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-0.0483 + 0.148i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 89.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + (23.0 + 70.8i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-131. - 95.6i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-59.3 - 182. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-84.9 + 261. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 396.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (2.04 - 6.30i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (374. + 271. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (242. + 747. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-103. + 75.5i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 - 594.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (531. - 386. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-296. - 913. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-359. - 261. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-794. + 577. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 - 1.37e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (410. + 298. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−13.40243834390147457797034682071, −12.70018972782195373497075044650, −11.03737248372080378202885974600, −10.11328633674573289930322944267, −9.052961658028529838558860114076, −8.091854034432395554305184394043, −6.34602563733036953332603378560, −5.03134259461060592397797176113, −2.25262563598737222176846359266, −0.938524226736821660002696694612,
2.66037174261872744818461823010, 4.53541226269063762648708079183, 6.46773740898629458424074870342, 7.37782184594509936642644755139, 9.271787251966494491069734661415, 9.480732613909860816305880202035, 10.65826836934851554775934212740, 12.35451041952114813672411111255, 13.44146292097144177146817430170, 14.75462289337345791080992346796
