L(s) = 1 | + 2.36·2-s + 2.84i·3-s + 3.57·4-s + 5-s + 6.72i·6-s + 0.816i·7-s + 3.71·8-s − 5.11·9-s + 2.36·10-s + (−3.22 − 0.764i)11-s + 10.1i·12-s − 0.847·13-s + 1.92i·14-s + 2.84i·15-s + 1.62·16-s + 4.07i·17-s + ⋯ |
L(s) = 1 | + 1.66·2-s + 1.64i·3-s + 1.78·4-s + 0.447·5-s + 2.74i·6-s + 0.308i·7-s + 1.31·8-s − 1.70·9-s + 0.746·10-s + (−0.973 − 0.230i)11-s + 2.93i·12-s − 0.234·13-s + 0.515i·14-s + 0.735i·15-s + 0.406·16-s + 0.987i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.131615375\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.131615375\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + (3.22 + 0.764i)T \) |
| 19 | \( 1 + (-2.17 - 3.77i)T \) |
good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 3 | \( 1 - 2.84iT - 3T^{2} \) |
| 7 | \( 1 - 0.816iT - 7T^{2} \) |
| 13 | \( 1 + 0.847T + 13T^{2} \) |
| 17 | \( 1 - 4.07iT - 17T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 31 | \( 1 + 5.91iT - 31T^{2} \) |
| 37 | \( 1 + 9.96iT - 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 5.14iT - 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 1.18iT - 53T^{2} \) |
| 59 | \( 1 - 1.07iT - 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 9.83iT - 67T^{2} \) |
| 71 | \( 1 + 4.42iT - 71T^{2} \) |
| 73 | \( 1 + 4.87iT - 73T^{2} \) |
| 79 | \( 1 + 5.76T + 79T^{2} \) |
| 83 | \( 1 + 11.9iT - 83T^{2} \) |
| 89 | \( 1 + 5.52iT - 89T^{2} \) |
| 97 | \( 1 - 13.9iT - 97T^{2} \) |
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
−10.51227021375166871148596515622, −9.525081202236842309234063325062, −8.703906518368934327030639045212, −7.50346992140843877703061179604, −6.09588086208846772552333714994, −5.59822721105682421635571126200, −4.89950069033853568860117772975, −4.12221137548614280606061343167, −3.24949166880804371499809954444, −2.42678991666562031190056236744,
1.14039744606239358203360440541, 2.66155136936236307265591974153, 2.88743938271776834320906264331, 4.78045836544641145377355290599, 5.21936113325379204151119703350, 6.34400244765960491599128792137, 6.92620591690648773639385294954, 7.47165439483657688106325194176, 8.543024462159007418179227674542, 9.794073429173681177147554290947