| L(s) = 1 | + (1.34 + 0.446i)2-s + (−0.402 + 0.972i)3-s + (1.60 + 1.19i)4-s + (−0.453 + 0.891i)5-s + (−0.975 + 1.12i)6-s + (0.454 + 1.89i)7-s + (1.61 + 2.32i)8-s + (1.33 + 1.33i)9-s + (−1.00 + 0.992i)10-s + (−0.828 + 0.970i)11-s + (−1.81 + 1.07i)12-s + (−1.31 − 2.14i)13-s + (−0.235 + 2.74i)14-s + (−0.683 − 0.800i)15-s + (1.12 + 3.83i)16-s + (−0.358 − 4.55i)17-s + ⋯ |
| L(s) = 1 | + (0.948 + 0.315i)2-s + (−0.232 + 0.561i)3-s + (0.800 + 0.599i)4-s + (−0.203 + 0.398i)5-s + (−0.398 + 0.459i)6-s + (0.171 + 0.716i)7-s + (0.570 + 0.821i)8-s + (0.445 + 0.445i)9-s + (−0.318 + 0.313i)10-s + (−0.249 + 0.292i)11-s + (−0.522 + 0.310i)12-s + (−0.364 − 0.594i)13-s + (−0.0629 + 0.733i)14-s + (−0.176 − 0.206i)15-s + (0.281 + 0.959i)16-s + (−0.0870 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21339 + 2.16756i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21339 + 2.16756i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.34 - 0.446i)T \) |
| 5 | \( 1 + (0.453 - 0.891i)T \) |
| 41 | \( 1 + (-0.337 - 6.39i)T \) |
| good | 3 | \( 1 + (0.402 - 0.972i)T + (-2.12 - 2.12i)T^{2} \) |
| 7 | \( 1 + (-0.454 - 1.89i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (0.828 - 0.970i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (1.31 + 2.14i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (0.358 + 4.55i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (1.85 + 1.13i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-2.39 - 1.74i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.0863 - 1.09i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (-0.500 + 1.53i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.272 - 0.839i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.787 - 4.97i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.80 + 7.51i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-7.70 - 0.606i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (2.13 - 2.93i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.11 - 7.04i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-3.83 + 3.27i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (-2.40 - 2.05i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-6.62 + 6.62i)T - 73iT^{2} \) |
| 79 | \( 1 + (15.3 + 6.36i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 13.3iT - 83T^{2} \) |
| 89 | \( 1 + (-16.9 + 4.06i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (-5.69 + 4.86i)T + (15.1 - 95.8i)T^{2} \) |
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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
−10.66500441668817139313357476572, −9.878026168479088822939505253515, −8.736293240578033052696656365178, −7.64052971169253589260102073279, −7.09251403325098565844779868136, −5.89858158579673446173299423200, −5.07060769991098761061558961730, −4.46562029690978619215660875756, −3.17804698166028992004805608531, −2.24414712378202813944474562908,
0.962762398070735234768474527147, 2.14091398477700493549794310352, 3.74902956832930770195466722408, 4.32940853078394404620719742219, 5.46912619543548552814683584700, 6.45033406151210160536381389082, 7.10296424056730656231381371439, 8.001514776945508428057258965410, 9.204514987052522393019981825141, 10.28905891720714162708654110438
