L(s) = 1 | + (1.61 + 1.17i)2-s + (−0.781 + 2.40i)3-s + (0.610 + 1.87i)4-s + (1.47 − 1.07i)5-s + (−4.07 + 2.96i)6-s + (−0.624 − 1.92i)7-s + (0.0147 − 0.0453i)8-s + (−2.74 − 1.99i)9-s + 3.64·10-s + (−2.99 + 1.41i)11-s − 4.99·12-s + (−0.809 − 0.587i)13-s + (1.24 − 3.83i)14-s + (1.42 + 4.39i)15-s + (3.27 − 2.37i)16-s + (1.34 − 0.975i)17-s + ⋯ |
L(s) = 1 | + (1.14 + 0.828i)2-s + (−0.451 + 1.38i)3-s + (0.305 + 0.939i)4-s + (0.660 − 0.480i)5-s + (−1.66 + 1.20i)6-s + (−0.236 − 0.726i)7-s + (0.00520 − 0.0160i)8-s + (−0.915 − 0.665i)9-s + 1.15·10-s + (−0.904 + 0.427i)11-s − 1.44·12-s + (−0.224 − 0.163i)13-s + (0.333 − 1.02i)14-s + (0.368 + 1.13i)15-s + (0.818 − 0.594i)16-s + (0.325 − 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12153 + 1.30536i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12153 + 1.30536i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (2.99 - 1.41i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
good | 2 | \( 1 + (-1.61 - 1.17i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.781 - 2.40i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (-1.47 + 1.07i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.624 + 1.92i)T + (-5.66 + 4.11i)T^{2} \) |
| 17 | \( 1 + (-1.34 + 0.975i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.610 - 1.87i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 3.93T + 23T^{2} \) |
| 29 | \( 1 + (2.68 + 8.25i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.14 - 1.55i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.429 - 1.32i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (2.86 - 8.83i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 + (1.90 - 5.85i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (9.98 + 7.25i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.23 + 3.81i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-12.5 + 9.14i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 6.46T + 67T^{2} \) |
| 71 | \( 1 + (2.88 - 2.09i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.140 - 0.431i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-3.90 - 2.83i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (6.13 - 4.45i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 2.42T + 89T^{2} \) |
| 97 | \( 1 + (-13.8 - 10.0i)T + (29.9 + 92.2i)T^{2} \) |
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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
−13.39673466444073672315559953304, −12.87929451691520890279888343325, −11.38492956363756019850284824789, −9.999618890235605082617960173654, −9.796069969653974892988603635370, −7.88369398183383774211262988229, −6.48219359035966900839264917444, −5.26037943019858314396334981716, −4.79323156293915179886524420206, −3.53000427333258338898116106298,
1.98142626856257195180650612984, 3.02479825304038861625980914620, 5.20699692199457465555116765609, 5.99764859764990921010779399491, 7.12222492494836737979713826833, 8.546162606640065723392220130109, 10.27011102182949079373486988654, 11.24012220983244057590470190143, 12.13063658012053619204097620501, 12.85923870562892711128554223652