L(s) = 1 | + (−2.13 − 0.656i)5-s + (1.13 − 2.38i)7-s + (−2.63 + 4.56i)11-s + 2.62i·13-s + (−0.362 − 0.209i)17-s + (1.63 + 2.83i)19-s + (6.77 − 3.91i)23-s + (4.13 + 2.80i)25-s + 4.27·29-s + (−1.63 + 2.83i)31-s + (−4 + 4.35i)35-s + (8.63 − 4.98i)37-s + 3.72·41-s − 2.15i·43-s + (5.63 − 3.25i)47-s + ⋯ |
L(s) = 1 | + (−0.955 − 0.293i)5-s + (0.429 − 0.902i)7-s + (−0.795 + 1.37i)11-s + 0.728i·13-s + (−0.0879 − 0.0507i)17-s + (0.375 + 0.650i)19-s + (1.41 − 0.815i)23-s + (0.827 + 0.561i)25-s + 0.793·29-s + (−0.294 + 0.509i)31-s + (−0.676 + 0.736i)35-s + (1.41 − 0.819i)37-s + 0.581·41-s − 0.327i·43-s + (0.822 − 0.474i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.341108617\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.341108617\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.13 + 0.656i)T \) |
| 7 | \( 1 + (-1.13 + 2.38i)T \) |
good | 11 | \( 1 + (2.63 - 4.56i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.62iT - 13T^{2} \) |
| 17 | \( 1 + (0.362 + 0.209i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.63 - 2.83i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.77 + 3.91i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 + (1.63 - 2.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.63 + 4.98i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 + 2.15iT - 43T^{2} \) |
| 47 | \( 1 + (-5.63 + 3.25i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.91 + 2.83i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.63 + 2.83i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.77 - 11.7i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.04 - 1.76i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 4.54T + 71T^{2} \) |
| 73 | \( 1 + (5.63 + 3.25i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.63 - 6.30i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.40iT - 83T^{2} \) |
| 89 | \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−9.764081610501987044782989442616, −8.835218351821493536117642934699, −7.943743769735318628797315722544, −7.32338072138089941322510743753, −6.75374886475519357756019139143, −5.18444937194707060705454569763, −4.54020982356149166871691173342, −3.83933341682787114833953574231, −2.44951994673837380359714826067, −0.987652687564147216651665321081,
0.76757521080139810802170603735, 2.75278272432819446987660799332, 3.20873576837898858020654706755, 4.61719668959566850266000957200, 5.42424324855726557040754554299, 6.24260177321851879161291704315, 7.43231684460042916142243478426, 8.060797251144186910784856588637, 8.668682954253171421567701503225, 9.537622529356355252472763923224