| L(s) = 1 | + (0.730 + 1.86i)2-s + (0.511 + 6.82i)3-s + (−2.93 + 2.72i)4-s + (−3.08 + 2.10i)5-s + (−12.3 + 5.93i)6-s + (15.0 + 10.8i)7-s + (−7.20 − 3.47i)8-s + (−19.6 + 2.95i)9-s + (−6.16 − 4.20i)10-s + (−35.0 − 5.27i)11-s + (−20.0 − 18.6i)12-s + (−5.31 + 6.66i)13-s + (−9.18 + 35.8i)14-s + (−15.9 − 19.9i)15-s + (1.19 − 15.9i)16-s + (1.31 − 0.406i)17-s + ⋯ |
| L(s) = 1 | + (0.258 + 0.658i)2-s + (0.0984 + 1.31i)3-s + (−0.366 + 0.340i)4-s + (−0.275 + 0.187i)5-s + (−0.839 + 0.404i)6-s + (0.811 + 0.584i)7-s + (−0.318 − 0.153i)8-s + (−0.726 + 0.109i)9-s + (−0.194 − 0.132i)10-s + (−0.959 − 0.144i)11-s + (−0.482 − 0.447i)12-s + (−0.113 + 0.142i)13-s + (−0.175 + 0.685i)14-s + (−0.273 − 0.343i)15-s + (0.0186 − 0.249i)16-s + (0.0188 − 0.00580i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 - 0.179i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.143276 + 1.58339i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143276 + 1.58339i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.730 - 1.86i)T \) |
| 7 | \( 1 + (-15.0 - 10.8i)T \) |
| good | 3 | \( 1 + (-0.511 - 6.82i)T + (-26.6 + 4.02i)T^{2} \) |
| 5 | \( 1 + (3.08 - 2.10i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (35.0 + 5.27i)T + (1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (5.31 - 6.66i)T + (-488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-1.31 + 0.406i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-57.5 + 99.7i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-36.3 - 11.2i)T + (1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (30.5 - 133. i)T + (-2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-106. - 185. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-124. - 115. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (-289. - 139. i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-144. + 69.4i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (223. + 569. i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-121. + 112. i)T + (1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-26.3 - 17.9i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (81.7 + 75.8i)T + (1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-17.7 - 30.7i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-44.6 - 195. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-325. + 828. i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (248. - 431. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (550. + 690. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-263. + 39.7i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 - 322.T + 9.12e5T^{2} \) |
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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
−14.30376283638273058824544130261, −13.09588704345750342912761256266, −11.63031194969247632702313885363, −10.71916369788368372126095476090, −9.449571984547124098517771363422, −8.492421411137780109520378485252, −7.24943128910317346452268409410, −5.38513690054196867246128431438, −4.69268642189149002644254332740, −3.11889757419403504164509841859,
0.886050590120911801292157193208, 2.35882282710667757156025490811, 4.33203442104749140100688658866, 5.88735678734523178653253184672, 7.60354608515917802787667467692, 8.057479376228443595455111556263, 9.880993828603846918975617101793, 11.07079147459517477132034770204, 12.07187116358012032781530093240, 12.86518117346967179830775011913
