| L(s) = 1 | + (−0.642 − 1.26i)2-s + (−0.0541 − 0.00857i)3-s + (−1.17 + 1.61i)4-s + (−1.39 + 4.80i)5-s + (0.0239 + 0.0737i)6-s + (−9.88 + 1.56i)7-s + (2.79 + 0.442i)8-s + (−8.55 − 2.78i)9-s + (6.94 − 1.32i)10-s + (0.380 + 10.9i)11-s + (0.0775 − 0.0775i)12-s + (1.29 − 0.659i)13-s + (8.32 + 11.4i)14-s + (0.116 − 0.248i)15-s + (−1.23 − 3.80i)16-s + (−0.376 − 0.191i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 − 0.630i)2-s + (−0.0180 − 0.00285i)3-s + (−0.293 + 0.404i)4-s + (−0.279 + 0.960i)5-s + (0.00399 + 0.0122i)6-s + (−1.41 + 0.223i)7-s + (0.349 + 0.0553i)8-s + (−0.950 − 0.308i)9-s + (0.694 − 0.132i)10-s + (0.0345 + 0.999i)11-s + (0.00646 − 0.00646i)12-s + (0.0996 − 0.0507i)13-s + (0.594 + 0.818i)14-s + (0.00778 − 0.0165i)15-s + (−0.0772 − 0.237i)16-s + (−0.0221 − 0.0112i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.259138 + 0.367893i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.259138 + 0.367893i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 + 1.26i)T \) |
| 5 | \( 1 + (1.39 - 4.80i)T \) |
| 11 | \( 1 + (-0.380 - 10.9i)T \) |
| good | 3 | \( 1 + (0.0541 + 0.00857i)T + (8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (9.88 - 1.56i)T + (46.6 - 15.1i)T^{2} \) |
| 13 | \( 1 + (-1.29 + 0.659i)T + (99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (0.376 + 0.191i)T + (169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-18.9 - 26.0i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (20.6 + 20.6i)T + 529iT^{2} \) |
| 29 | \( 1 + (-22.6 + 31.2i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (11.3 - 34.9i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (26.0 - 4.12i)T + (1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (18.5 - 13.4i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-8.47 - 8.47i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-6.02 + 38.0i)T + (-2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-69.5 + 35.4i)T + (1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (67.8 - 93.3i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-19.5 - 60.1i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (49.6 - 49.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + (-0.774 - 2.38i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-15.3 - 97.1i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-83.9 - 27.2i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (37.1 - 72.9i)T + (-4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + 104. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (27.0 + 53.1i)T + (-5.53e3 + 7.61e3i)T^{2} \) |
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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
−13.70389513666402307800728375798, −12.28208318689042974679636602935, −11.86681684579358420438218048250, −10.32517898292701987541311881303, −9.842416010067482011102361090787, −8.474996522622538649736686159376, −7.08837284473388217827572824507, −5.96290166732783215967506181386, −3.74475912003227523645804058419, −2.67207320700937310870738761856,
0.35048941445852111278311988656, 3.39936073388141298512128890523, 5.22554981015088329663738473057, 6.25723364278120888124590773201, 7.66071298658684103172414427901, 8.840129243154284208980435653814, 9.489843818910005027252722553551, 10.96215913957426687852465210633, 12.12689550644072933322082348775, 13.43015258148094298579679125878
