L(s) = 1 | + (−0.309 + 1.37i)2-s + (−1.80 − 0.853i)4-s + i·5-s + (2.64 − 0.0785i)7-s + (1.73 − 2.23i)8-s + (−1.37 − 0.309i)10-s + 0.987i·11-s − 4.69i·13-s + (−0.709 + 3.67i)14-s + (2.54 + 3.08i)16-s + 3.93i·17-s − 0.223·19-s + (0.853 − 1.80i)20-s + (−1.36 − 0.305i)22-s − 5.88i·23-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.975i)2-s + (−0.904 − 0.426i)4-s + 0.447i·5-s + (0.999 − 0.0296i)7-s + (0.614 − 0.789i)8-s + (−0.436 − 0.0977i)10-s + 0.297i·11-s − 1.30i·13-s + (−0.189 + 0.981i)14-s + (0.635 + 0.771i)16-s + 0.954i·17-s − 0.0513·19-s + (0.190 − 0.404i)20-s + (−0.290 − 0.0650i)22-s − 1.22i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.530143043\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.530143043\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 1.37i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-2.64 + 0.0785i)T \) |
good | 11 | \( 1 - 0.987iT - 11T^{2} \) |
| 13 | \( 1 + 4.69iT - 13T^{2} \) |
| 17 | \( 1 - 3.93iT - 17T^{2} \) |
| 19 | \( 1 + 0.223T + 19T^{2} \) |
| 23 | \( 1 + 5.88iT - 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 - 8.26T + 37T^{2} \) |
| 41 | \( 1 + 7.34iT - 41T^{2} \) |
| 43 | \( 1 - 4.32iT - 43T^{2} \) |
| 47 | \( 1 + 2.40T + 47T^{2} \) |
| 53 | \( 1 - 8.35T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 4.93iT - 61T^{2} \) |
| 67 | \( 1 - 7.84iT - 67T^{2} \) |
| 71 | \( 1 - 8.49iT - 71T^{2} \) |
| 73 | \( 1 - 14.4iT - 73T^{2} \) |
| 79 | \( 1 - 11.5iT - 79T^{2} \) |
| 83 | \( 1 - 1.67T + 83T^{2} \) |
| 89 | \( 1 - 0.493iT - 89T^{2} \) |
| 97 | \( 1 - 4.31iT - 97T^{2} \) |
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
−9.943419379167342893274577033740, −8.571589324798018692715031201021, −8.252707324552361967326637646739, −7.47651594414680013786290711433, −6.54200169379110317412422745527, −5.79433737330847650388696201708, −4.84780358309226999783570101545, −4.08849066598894875608971896437, −2.62033409450870504969865605161, −1.00081360878366170367159832714,
1.01729481619968072130910797233, 2.04085604592119426002799405134, 3.20522166075585936457291005983, 4.55697020465057826673906240575, 4.76362202852140939377022754327, 6.07702044905927435891794123701, 7.39284191823174894036479014231, 8.114733059004189254711575107080, 8.938123535768920815603723135893, 9.462043130005301082163572610122