L(s) = 1 | + (−0.415 − 0.909i)2-s + (1.10 + 0.708i)3-s + (−0.654 + 0.755i)4-s + (−0.142 − 0.989i)5-s + (0.186 − 1.29i)6-s + (−0.698 − 1.53i)7-s + (0.959 + 0.281i)8-s + (0.297 + 0.650i)9-s + (−0.841 + 0.540i)10-s + (−1.25 + 0.368i)12-s + (−1.10 + 1.27i)14-s + (0.544 − 1.19i)15-s + (−0.142 − 0.989i)16-s + (0.468 − 0.540i)18-s + (0.841 + 0.540i)20-s + (0.313 − 2.18i)21-s + ⋯ |
L(s) = 1 | + (−0.415 − 0.909i)2-s + (1.10 + 0.708i)3-s + (−0.654 + 0.755i)4-s + (−0.142 − 0.989i)5-s + (0.186 − 1.29i)6-s + (−0.698 − 1.53i)7-s + (0.959 + 0.281i)8-s + (0.297 + 0.650i)9-s + (−0.841 + 0.540i)10-s + (−1.25 + 0.368i)12-s + (−1.10 + 1.27i)14-s + (0.544 − 1.19i)15-s + (−0.142 − 0.989i)16-s + (0.468 − 0.540i)18-s + (0.841 + 0.540i)20-s + (0.313 − 2.18i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9961838149\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9961838149\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.415 + 0.909i)T \) |
| 5 | \( 1 + (0.142 + 0.989i)T \) |
| 67 | \( 1 + (-0.654 + 0.755i)T \) |
good | 3 | \( 1 + (-1.10 - 0.708i)T + (0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (0.698 + 1.53i)T + (-0.654 + 0.755i)T^{2} \) |
| 11 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 13 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 17 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 19 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 23 | \( 1 + (1.68 + 1.08i)T + (0.415 + 0.909i)T^{2} \) |
| 29 | \( 1 - 1.68T + T^{2} \) |
| 31 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 43 | \( 1 + (0.857 + 0.989i)T + (-0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 + (-1.61 - 1.03i)T + (0.415 + 0.909i)T^{2} \) |
| 53 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 61 | \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \) |
| 71 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 73 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 79 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 83 | \( 1 + (-0.118 - 0.822i)T + (-0.959 + 0.281i)T^{2} \) |
| 89 | \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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
−9.683236585880899134082892783024, −8.913651553011596806111418280971, −8.180286786410928059597640654978, −7.66353616013503381656547498414, −6.37213576698966090165840412931, −4.65298040746243484013953457752, −4.17574342543011710334705388280, −3.51404890565090870566505812981, −2.43492863134600800633385709667, −0.868054638299173842371867989291,
2.01047483585881177405901166360, 2.79334091630481135564566988509, 3.88267779245881458609974455984, 5.49350496762291365410340434107, 6.16389050968073596787363062038, 6.91275668637051797485612141722, 7.70661919881539050429568047214, 8.345691859803245949595039562093, 9.019670361654395100707754341353, 9.738410079525235117771735994661