| L(s) = 1 | + (0.939 + 0.342i)2-s + (1.69 − 0.377i)3-s + (0.766 + 0.642i)4-s + (−0.173 + 0.984i)5-s + (1.71 + 0.223i)6-s + (−0.268 + 0.225i)7-s + (0.500 + 0.866i)8-s + (2.71 − 1.27i)9-s + (−0.5 + 0.866i)10-s + (−0.347 − 1.97i)11-s + (1.53 + 0.797i)12-s + (−5.74 + 2.09i)13-s + (−0.329 + 0.119i)14-s + (0.0783 + 1.73i)15-s + (0.173 + 0.984i)16-s + (2.09 − 3.62i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.975 − 0.217i)3-s + (0.383 + 0.321i)4-s + (−0.0776 + 0.440i)5-s + (0.701 + 0.0911i)6-s + (−0.101 + 0.0851i)7-s + (0.176 + 0.306i)8-s + (0.904 − 0.425i)9-s + (−0.158 + 0.273i)10-s + (−0.104 − 0.594i)11-s + (0.443 + 0.230i)12-s + (−1.59 + 0.580i)13-s + (−0.0880 + 0.0320i)14-s + (0.0202 + 0.446i)15-s + (0.0434 + 0.246i)16-s + (0.508 − 0.880i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.26675 + 0.422300i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.26675 + 0.422300i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (-1.69 + 0.377i)T \) |
| 5 | \( 1 + (0.173 - 0.984i)T \) |
| good | 7 | \( 1 + (0.268 - 0.225i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.347 + 1.97i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (5.74 - 2.09i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.09 + 3.62i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.788 - 1.36i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0803 + 0.0674i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (5.69 + 2.07i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (3.88 + 3.25i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (0.735 - 1.27i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.64 + 2.41i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.26 - 7.18i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.07 + 3.41i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + (0.470 - 2.67i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.67 + 3.08i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (14.5 - 5.29i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.90 + 3.29i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.483 - 0.836i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.97 - 2.53i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-9.07 - 3.30i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (8.16 + 14.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.81 + 10.2i)T + (-91.1 + 33.1i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.19300129849574135332946358547, −11.25427763415053676298299722803, −9.891275243231368788035275066493, −9.143775970132099011018350263903, −7.68999346200800912480612653126, −7.30659052619311526429063543799, −5.99840258882739844112116287324, −4.61233192055774419210800843077, −3.34649165333133609042858795578, −2.32665789383405558822842090637,
2.01048455141726090006842815094, 3.31405682745239643461131508200, 4.48756825487258427598975875452, 5.43734103112387255215749113749, 7.14997029564564636989292899819, 7.85016205574350586628614672138, 9.172927636439677952733960950498, 9.939888528480943172872145205746, 10.82194010229408236364855177877, 12.39211615347213701949122626192
