L(s) = 1 | + (0.981 + 0.189i)3-s + (0.959 − 0.281i)4-s + (−0.580 + 0.814i)7-s + (0.928 + 0.371i)9-s + (0.995 − 0.0950i)12-s + (0.283 − 0.0270i)13-s + (0.841 − 0.540i)16-s + (−1.87 − 0.268i)19-s + (−0.723 + 0.690i)21-s + (−0.995 + 0.0950i)25-s + (0.841 + 0.540i)27-s + (−0.327 + 0.945i)28-s + (−0.698 − 1.53i)31-s + (0.995 + 0.0950i)36-s + 1.16·37-s + ⋯ |
L(s) = 1 | + (0.981 + 0.189i)3-s + (0.959 − 0.281i)4-s + (−0.580 + 0.814i)7-s + (0.928 + 0.371i)9-s + (0.995 − 0.0950i)12-s + (0.283 − 0.0270i)13-s + (0.841 − 0.540i)16-s + (−1.87 − 0.268i)19-s + (−0.723 + 0.690i)21-s + (−0.995 + 0.0950i)25-s + (0.841 + 0.540i)27-s + (−0.327 + 0.945i)28-s + (−0.698 − 1.53i)31-s + (0.995 + 0.0950i)36-s + 1.16·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.740788288\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.740788288\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.981 - 0.189i)T \) |
| 7 | \( 1 + (0.580 - 0.814i)T \) |
| 67 | \( 1 + (0.580 + 0.814i)T \) |
good | 2 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 5 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 11 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 13 | \( 1 + (-0.283 + 0.0270i)T + (0.981 - 0.189i)T^{2} \) |
| 17 | \( 1 + (0.0475 - 0.998i)T^{2} \) |
| 19 | \( 1 + (1.87 + 0.268i)T + (0.959 + 0.281i)T^{2} \) |
| 23 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.698 + 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
| 37 | \( 1 - 1.16T + T^{2} \) |
| 41 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 43 | \( 1 + (0.512 - 1.74i)T + (-0.841 - 0.540i)T^{2} \) |
| 47 | \( 1 + (0.928 + 0.371i)T^{2} \) |
| 53 | \( 1 + (0.0475 + 0.998i)T^{2} \) |
| 59 | \( 1 + (-0.327 - 0.945i)T^{2} \) |
| 61 | \( 1 + (-0.550 - 0.353i)T + (0.415 + 0.909i)T^{2} \) |
| 71 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 73 | \( 1 + (0.915 + 1.77i)T + (-0.580 + 0.814i)T^{2} \) |
| 79 | \( 1 + (0.154 + 1.62i)T + (-0.981 + 0.189i)T^{2} \) |
| 83 | \( 1 + (0.580 + 0.814i)T^{2} \) |
| 89 | \( 1 + (0.928 - 0.371i)T^{2} \) |
| 97 | \( 1 + (-0.981 - 1.70i)T + (-0.5 + 0.866i)T^{2} \) |
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
−9.679239852315826427677054244793, −9.073052185129314283952796735921, −8.147752033314469301319883763247, −7.50659883240910368521310768294, −6.37283243864299123029433400751, −5.98741344586093731762677011012, −4.59601184212343384923929890670, −3.54174048821990545433198042873, −2.55466128502101915617744735658, −1.90759502706131066113973381323,
1.61995979299475932804378228882, 2.59764643311395038830793457544, 3.62223116961172944337116143812, 4.19971468982550385574794322258, 5.86944770628810252383501161654, 6.78359327232243857438513941602, 7.16878729697214249886953182944, 8.157770471474146066149830907125, 8.687005828737489226596843480091, 9.822164947040527159020669739213