L(s) = 1 | + (−1.18 − 2.75i)3-s + (0.00985 − 0.00985i)5-s + 6.42i·7-s + (−6.19 + 6.53i)9-s + (9.07 − 9.07i)11-s + (−12.6 + 12.6i)13-s + (−0.0388 − 0.0154i)15-s + 19.0i·17-s + (−2.07 + 2.07i)19-s + (17.7 − 7.61i)21-s − 19.5·23-s + 24.9i·25-s + (25.3 + 9.32i)27-s + (11.1 + 11.1i)29-s + 59.9·31-s + ⋯ |
L(s) = 1 | + (−0.395 − 0.918i)3-s + (0.00197 − 0.00197i)5-s + 0.917i·7-s + (−0.687 + 0.725i)9-s + (0.824 − 0.824i)11-s + (−0.969 + 0.969i)13-s + (−0.00259 − 0.00103i)15-s + 1.11i·17-s + (−0.109 + 0.109i)19-s + (0.842 − 0.362i)21-s − 0.850·23-s + 0.999i·25-s + (0.938 + 0.345i)27-s + (0.385 + 0.385i)29-s + 1.93·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.981265 + 0.508105i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.981265 + 0.508105i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.18 + 2.75i)T \) |
good | 5 | \( 1 + (-0.00985 + 0.00985i)T - 25iT^{2} \) |
| 7 | \( 1 - 6.42iT - 49T^{2} \) |
| 11 | \( 1 + (-9.07 + 9.07i)T - 121iT^{2} \) |
| 13 | \( 1 + (12.6 - 12.6i)T - 169iT^{2} \) |
| 17 | \( 1 - 19.0iT - 289T^{2} \) |
| 19 | \( 1 + (2.07 - 2.07i)T - 361iT^{2} \) |
| 23 | \( 1 + 19.5T + 529T^{2} \) |
| 29 | \( 1 + (-11.1 - 11.1i)T + 841iT^{2} \) |
| 31 | \( 1 - 59.9T + 961T^{2} \) |
| 37 | \( 1 + (9.32 + 9.32i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 47.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-24.1 - 24.1i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 6.29iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (20.6 - 20.6i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (60.3 - 60.3i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (48.0 - 48.0i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (23.7 - 23.7i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 13.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 31.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 47.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + (70.3 + 70.3i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 95.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 61.6T + 9.40e3T^{2} \) |
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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
−11.61598483340559198910914275712, −10.51057377911119748427189563538, −9.220628517035075231983097337665, −8.493960609544206362674767922285, −7.46463248436064743382712050726, −6.33027439488424401961286242606, −5.81247240808412952359591472491, −4.40203811293161631949592111636, −2.72195603313904103465738550521, −1.46641592552183654953392966279,
0.53757781143296179816489910721, 2.79180750832344804208462705653, 4.22633287772968513621241790816, 4.79730660846037572202828378476, 6.14820481142849072022362907814, 7.15725366004801434938852224979, 8.207487660807850352371050991401, 9.604916931772454022012141344128, 9.956134555969956380527018793911, 10.81695581336087573422264706946