| L(s) = 1 | + (−0.347 − 1.96i)2-s + (1.09 − 6.19i)3-s + (−3.75 + 1.36i)4-s + (−4.09 − 3.43i)5-s − 12.5·6-s + (−9.53 − 8.00i)7-s + (4 + 6.92i)8-s + (−11.8 − 4.31i)9-s + (−5.34 + 9.25i)10-s + (1.97 + 3.42i)11-s + (4.37 + 24.7i)12-s + (−17.2 + 6.29i)13-s + (−12.4 + 21.5i)14-s + (−25.7 + 21.6i)15-s + (12.2 − 10.2i)16-s + (−38.2 − 13.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.210 − 1.19i)3-s + (−0.469 + 0.171i)4-s + (−0.366 − 0.307i)5-s − 0.856·6-s + (−0.514 − 0.432i)7-s + (0.176 + 0.306i)8-s + (−0.438 − 0.159i)9-s + (−0.169 + 0.292i)10-s + (0.0541 + 0.0938i)11-s + (0.105 + 0.596i)12-s + (−0.368 + 0.134i)13-s + (−0.237 + 0.411i)14-s + (−0.443 + 0.372i)15-s + (0.191 − 0.160i)16-s + (−0.545 − 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0159i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00836722 + 1.04597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00836722 + 1.04597i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.347 + 1.96i)T \) |
| 37 | \( 1 + (-200. + 102. i)T \) |
| good | 3 | \( 1 + (-1.09 + 6.19i)T + (-25.3 - 9.23i)T^{2} \) |
| 5 | \( 1 + (4.09 + 3.43i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (9.53 + 8.00i)T + (59.5 + 337. i)T^{2} \) |
| 11 | \( 1 + (-1.97 - 3.42i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (17.2 - 6.29i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (38.2 + 13.9i)T + (3.76e3 + 3.15e3i)T^{2} \) |
| 19 | \( 1 + (-10.3 + 58.9i)T + (-6.44e3 - 2.34e3i)T^{2} \) |
| 23 | \( 1 + (-33.3 + 57.7i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-7.55 - 13.0i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 186.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (115. - 42.0i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 - 257.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-87.2 + 151. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (118. - 99.2i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (407. - 342. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (463. - 168. i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-401. - 336. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (139. - 789. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 - 416.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-540. - 453. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (308. + 112. i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (-474. + 398. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-811. + 1.40e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.25250169848798242716002361080, −12.51852601542731203003909574671, −11.58568543436543449355291271156, −10.21249745695182115424823812999, −8.881821883229688308159211705998, −7.68116979092648746455315312843, −6.59605260543961123123119799213, −4.44176916324185018783363358525, −2.53814357870641703998419341870, −0.70321129335847625255275801699,
3.37709656872609851123866781186, 4.74264596332475282078439823756, 6.21999372928893429963743859083, 7.72626967469530677428986047220, 9.094569811946749741019235682392, 9.866861975601067277130826909630, 11.03638932846862880630211836960, 12.51502379362635757624077607559, 13.89520181264056032444026287127, 15.11279762493460988239733513146
