| L(s) = 1 | + (1.10 + 2.81i)2-s + (0.393 + 5.24i)3-s + (−0.820 + 0.761i)4-s + (−13.6 + 9.27i)5-s + (−14.3 + 6.89i)6-s + (6.02 − 17.5i)7-s + (18.7 + 9.01i)8-s + (−0.701 + 0.105i)9-s + (−41.0 − 28.0i)10-s + (12.7 + 1.92i)11-s + (−4.31 − 4.00i)12-s + (−52.0 + 65.2i)13-s + (55.8 − 2.39i)14-s + (−54.0 − 67.7i)15-s + (−5.35 + 71.5i)16-s + (119. − 36.7i)17-s + ⋯ |
| L(s) = 1 | + (0.390 + 0.993i)2-s + (0.0757 + 1.01i)3-s + (−0.102 + 0.0951i)4-s + (−1.21 + 0.829i)5-s + (−0.974 + 0.469i)6-s + (0.325 − 0.945i)7-s + (0.827 + 0.398i)8-s + (−0.0259 + 0.00391i)9-s + (−1.29 − 0.886i)10-s + (0.350 + 0.0528i)11-s + (−0.103 − 0.0964i)12-s + (−1.11 + 1.39i)13-s + (1.06 − 0.0457i)14-s + (−0.930 − 1.16i)15-s + (−0.0837 + 1.11i)16-s + (1.70 − 0.524i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.613072 + 1.46944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.613072 + 1.46944i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-6.02 + 17.5i)T \) |
| good | 2 | \( 1 + (-1.10 - 2.81i)T + (-5.86 + 5.44i)T^{2} \) |
| 3 | \( 1 + (-0.393 - 5.24i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (13.6 - 9.27i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (-12.7 - 1.92i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (52.0 - 65.2i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-119. + 36.7i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-47.4 + 82.2i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-11.9 - 3.69i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-17.8 + 78.2i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (99.4 + 172. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (168. + 155. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (213. + 103. i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-111. + 53.8i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-65.4 - 166. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (59.9 - 55.6i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (274. + 186. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-196. - 182. i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (177. + 307. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-127. - 559. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-263. + 670. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (145. - 252. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-232. - 291. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (171. - 25.8i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 - 310.T + 9.12e5T^{2} \) |
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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
−15.45634270579848627670207533299, −14.60978646456506550903728572048, −14.03781139449033211858245712134, −11.80070588563495950839179407260, −10.84445221364589596721390726589, −9.646394996090220401076998599903, −7.52606455945054570083117159084, −7.06510476657343879807637978758, −4.86276214522338286870987272883, −3.83802520091011241088817741099,
1.32661803939203161952727785875, 3.30787662763965413620248961210, 5.15934418614511948865538577849, 7.51168673922958730197182053677, 8.194004461626856737485464763055, 10.19009700734513451072841680195, 11.94408690483542130115124060114, 12.23395306963559433976872629048, 12.81447431368404280134117217084, 14.48740223548461313253231999468
