L(s) = 1 | + (0.578 − 1.29i)2-s + (−0.751 − 1.56i)3-s + (−1.33 − 1.49i)4-s + (−2.44 + 0.0670i)6-s − 4.28i·7-s + (−2.69 + 0.852i)8-s + (−1.86 + 2.34i)9-s − 2.44i·11-s + (−1.33 + 3.19i)12-s + 2.71i·13-s + (−5.53 − 2.48i)14-s + (−0.460 + 3.97i)16-s − 1.16i·17-s + (1.94 + 3.77i)18-s + 6.05·19-s + ⋯ |
L(s) = 1 | + (0.409 − 0.912i)2-s + (−0.433 − 0.900i)3-s + (−0.665 − 0.746i)4-s + (−0.999 + 0.0273i)6-s − 1.61i·7-s + (−0.953 + 0.301i)8-s + (−0.623 + 0.781i)9-s − 0.737i·11-s + (−0.384 + 0.923i)12-s + 0.752i·13-s + (−1.47 − 0.662i)14-s + (−0.115 + 0.993i)16-s − 0.282i·17-s + (0.458 + 0.888i)18-s + 1.38·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.420403 + 0.972979i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.420403 + 0.972979i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.578 + 1.29i)T \) |
| 3 | \( 1 + (0.751 + 1.56i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4.28iT - 7T^{2} \) |
| 11 | \( 1 + 2.44iT - 11T^{2} \) |
| 13 | \( 1 - 2.71iT - 13T^{2} \) |
| 17 | \( 1 + 1.16iT - 17T^{2} \) |
| 19 | \( 1 - 6.05T + 19T^{2} \) |
| 23 | \( 1 + 7.55T + 23T^{2} \) |
| 29 | \( 1 + 0.733T + 29T^{2} \) |
| 31 | \( 1 - 0.469iT - 31T^{2} \) |
| 37 | \( 1 - 1.36iT - 37T^{2} \) |
| 41 | \( 1 + 4.69iT - 41T^{2} \) |
| 43 | \( 1 - 1.50T + 43T^{2} \) |
| 47 | \( 1 + 4.07T + 47T^{2} \) |
| 53 | \( 1 - 1.00T + 53T^{2} \) |
| 59 | \( 1 - 1.63iT - 59T^{2} \) |
| 61 | \( 1 + 10.9iT - 61T^{2} \) |
| 67 | \( 1 - 9.97T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 + 3.61iT - 79T^{2} \) |
| 83 | \( 1 + 5.45iT - 83T^{2} \) |
| 89 | \( 1 + 7.75iT - 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−10.36617415974224399428174712466, −9.557518425589973976156984181412, −8.278404874574413310816219147560, −7.35809458618937929998564261646, −6.42994808469845598077469701287, −5.42732586858388636243353874546, −4.29704776578315981266884898742, −3.27525917407241789212454813018, −1.72436461101113191779909007693, −0.55472633370798204261219123808,
2.73355695209280711533412326181, 3.88644846360659373212068104349, 5.08251771785076511806712754474, 5.60841577832611966380768650123, 6.37571532292705626397026645625, 7.73574758130793073542119638416, 8.561036886170093711082587398681, 9.467281119781650282549337377029, 9.999631980647139861477521606206, 11.45618830406320669208538835860