L(s) = 1 | + (0.437 − 1.34i)2-s + (−1.61 − 1.17i)4-s − 4.29i·5-s − 7-s + (−2.28 + 1.66i)8-s + (−5.78 − 1.87i)10-s − 1.60i·11-s − 6.02i·13-s + (−0.437 + 1.34i)14-s + (1.23 + 3.80i)16-s + 3.16·17-s − 3.07i·19-s + (−5.05 + 6.95i)20-s + (−2.16 − 0.702i)22-s + 5.95·23-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s − 1.92i·5-s − 0.377·7-s + (−0.809 + 0.587i)8-s + (−1.82 − 0.593i)10-s − 0.484i·11-s − 1.67i·13-s + (−0.116 + 0.359i)14-s + (0.309 + 0.951i)16-s + 0.766·17-s − 0.706i·19-s + (−1.12 + 1.55i)20-s + (−0.460 − 0.149i)22-s + 1.24·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.373123886\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.373123886\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.437 + 1.34i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 4.29iT - 5T^{2} \) |
| 11 | \( 1 + 1.60iT - 11T^{2} \) |
| 13 | \( 1 + 6.02iT - 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 + 3.07iT - 19T^{2} \) |
| 23 | \( 1 - 5.95T + 23T^{2} \) |
| 29 | \( 1 - 3.53iT - 29T^{2} \) |
| 31 | \( 1 + 2.08T + 31T^{2} \) |
| 37 | \( 1 + 4.48iT - 37T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 - 12.2iT - 43T^{2} \) |
| 47 | \( 1 - 3.82T + 47T^{2} \) |
| 53 | \( 1 - 8.63iT - 53T^{2} \) |
| 59 | \( 1 - 1.55iT - 59T^{2} \) |
| 61 | \( 1 + 3.64iT - 61T^{2} \) |
| 67 | \( 1 + 0.772iT - 67T^{2} \) |
| 71 | \( 1 - 7.36T + 71T^{2} \) |
| 73 | \( 1 - 1.32T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 7.08iT - 83T^{2} \) |
| 89 | \( 1 - 9.73T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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
−9.092101109664038300245819010043, −8.444822975469514238225474096619, −7.67802980453082868107254071249, −6.05289329782622051709544955066, −5.26303353741594227153438477566, −4.89479242452403735759379746667, −3.70212600793113283238043058010, −2.85639394310541416172223504252, −1.27837663822684134835167051081, −0.54065523611205638315147407808,
2.20752845585234580651790000974, 3.38011493773451905917502112878, 3.94305353379734585830018665644, 5.21200616976680870270175273503, 6.22634593936934268068478182078, 6.86095856234856478781996588992, 7.17682019889159994812490732652, 8.124368104133044917630825022325, 9.240949416365389916478329893462, 9.875872192269302492105915310279