| L(s) = 1 | + (2.71 − 8.34i)2-s + (−5.61 − 4.07i)3-s + (−36.3 − 26.4i)4-s + (−55.5 − 6.03i)5-s + (−49.2 + 35.7i)6-s + 102.·7-s + (−91.9 + 66.8i)8-s + (−60.2 − 185. i)9-s + (−201. + 447. i)10-s + (−24.8 + 76.6i)11-s + (96.3 + 296. i)12-s + (−82.4 − 253. i)13-s + (277. − 853. i)14-s + (287. + 260. i)15-s + (−136. − 419. i)16-s + (653. − 474. i)17-s + ⋯ |
| L(s) = 1 | + (0.479 − 1.47i)2-s + (−0.359 − 0.261i)3-s + (−1.13 − 0.825i)4-s + (−0.994 − 0.107i)5-s + (−0.558 + 0.405i)6-s + 0.788·7-s + (−0.508 + 0.369i)8-s + (−0.247 − 0.762i)9-s + (−0.635 + 1.41i)10-s + (−0.0620 + 0.190i)11-s + (0.193 + 0.594i)12-s + (−0.135 − 0.416i)13-s + (0.378 − 1.16i)14-s + (0.329 + 0.298i)15-s + (−0.133 − 0.409i)16-s + (0.548 − 0.398i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0349i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0234126 - 1.33983i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0234126 - 1.33983i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (55.5 + 6.03i)T \) |
| good | 2 | \( 1 + (-2.71 + 8.34i)T + (-25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (5.61 + 4.07i)T + (75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 102.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (24.8 - 76.6i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (82.4 + 253. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-653. + 474. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-2.24e3 + 1.62e3i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (1.10e3 - 3.40e3i)T + (-5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (436. + 316. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.42e3 + 1.03e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-949. - 2.92e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-5.28e3 - 1.62e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.61e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.41e4 + 1.03e4i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.22e4 - 8.93e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.14e4 + 3.51e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.68e4 + 5.17e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-3.28e3 + 2.38e3i)T + (4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-1.85e4 - 1.35e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (2.19e4 - 6.74e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (4.47e4 + 3.25e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-4.53e4 + 3.29e4i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-3.36e4 + 1.03e5i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-3.82e4 - 2.78e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−15.74944679634123699549083550143, −14.35803957471820200891495577543, −12.90039357346498802657133994274, −11.66966170634323006382385190947, −11.40777659625588588208187720670, −9.579793123280283745290756523597, −7.59093472705293022367823032687, −4.99681163500646674443736944007, −3.31846682905871295012404793680, −0.904410229088581970310271338655,
4.32963113153506543187594829041, 5.62659114392538077472824637446, 7.46004386853456988683462659503, 8.315398886756559286532859367609, 10.77125357935925059963118889817, 12.12306522602354603043770824691, 14.00453937410548486591408479562, 14.73340252423097440106685508134, 16.11896999500835967888579723938, 16.52052871735676487835297245500
