L(s) = 1 | + (−1.36 + 2.36i)2-s + (−0.673 − 1.16i)3-s + (−2.71 − 4.69i)4-s + (1.09 − 1.89i)5-s + 3.66·6-s + 9.33·8-s + (0.593 − 1.02i)9-s + (2.98 + 5.16i)10-s + (0.524 + 0.907i)11-s + (−3.65 + 6.32i)12-s − 13-s − 2.94·15-s + (−7.29 + 12.6i)16-s + (−2.64 − 4.58i)17-s + (1.61 + 2.80i)18-s + (0.378 − 0.655i)19-s + ⋯ |
L(s) = 1 | + (−0.963 + 1.66i)2-s + (−0.388 − 0.673i)3-s + (−1.35 − 2.34i)4-s + (0.489 − 0.847i)5-s + 1.49·6-s + 3.30·8-s + (0.197 − 0.342i)9-s + (0.942 + 1.63i)10-s + (0.158 + 0.273i)11-s + (−1.05 + 1.82i)12-s − 0.277·13-s − 0.760·15-s + (−1.82 + 3.16i)16-s + (−0.641 − 1.11i)17-s + (0.381 + 0.660i)18-s + (0.0868 − 0.150i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.447657 - 0.297773i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.447657 - 0.297773i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + (1.36 - 2.36i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.673 + 1.16i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.09 + 1.89i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.524 - 0.907i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2.64 + 4.58i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.378 + 0.655i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.326 - 0.566i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.10T + 29T^{2} \) |
| 31 | \( 1 + (-0.513 - 0.890i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.44 + 9.43i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7.32T + 41T^{2} \) |
| 43 | \( 1 - 0.887T + 43T^{2} \) |
| 47 | \( 1 + (-1.16 + 2.02i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.44 + 4.23i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.524 + 0.907i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.24 - 10.8i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.23 + 3.87i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6.60T + 71T^{2} \) |
| 73 | \( 1 + (4.14 + 7.17i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.07 - 1.85i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 + (2.88 - 4.99i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.88T + 97T^{2} \) |
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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
−9.832998236907154685262966120607, −9.306933854465863408556833824703, −8.684527519997072663864102586542, −7.48726941882056991968612698939, −7.05665322126163281527216324187, −6.11116971821085029702495207443, −5.36499972964153695003218330480, −4.47938349446214616066162659524, −1.66182544439942554344559947374, −0.44028680681346260899922833916,
1.70217183328077112493949348873, 2.76663578094692919485230202755, 3.84674534042961743953037604217, 4.77188941934893425910890704959, 6.30154255488886456688419677726, 7.60001356097896440247744695087, 8.485839190671764509081686824462, 9.462342276217135799663855220641, 10.15075693554016048993210311364, 10.61390199376315823979350932954