L(s) = 1 | + (−1 + 2i)5-s − 3i·9-s + (1 − i)13-s + (3 − 3i)17-s + (−3 − 4i)25-s + 4·29-s + (7 + 7i)37-s + 8·41-s + (6 + 3i)45-s − 7i·49-s + (9 − 9i)53-s + 12i·61-s + (1 + 3i)65-s + (11 + 11i)73-s − 9·81-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.894i)5-s − i·9-s + (0.277 − 0.277i)13-s + (0.727 − 0.727i)17-s + (−0.600 − 0.800i)25-s + 0.742·29-s + (1.15 + 1.15i)37-s + 1.24·41-s + (0.894 + 0.447i)45-s − i·49-s + (1.23 − 1.23i)53-s + 1.53i·61-s + (0.124 + 0.372i)65-s + (1.28 + 1.28i)73-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.516198183\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.516198183\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
good | 3 | \( 1 + 3iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3 + 3i)T - 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (-7 - 7i)T + 37iT^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (-9 + 9i)T - 53iT^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 12iT - 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-11 - 11i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (-13 + 13i)T - 97iT^{2} \) |
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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
−9.825958827483972887153904020253, −8.800093670867458467264138813393, −7.955683167132138330383180203104, −7.12374153116788240812881369957, −6.43879918036070445624435376550, −5.58255744935582529467512340996, −4.31530874717417801948092227188, −3.41917159381795126682362195226, −2.63468761777757803659109401171, −0.820320506913134344902456481871,
1.07560883208194430273964421974, 2.35932758148223026856062009519, 3.77087942110642421376506493881, 4.55994791326550828512608548662, 5.42280719842764906931291849703, 6.26431745713657614381074924810, 7.62047579371549581888179827572, 7.938343535944210077954830740999, 8.868856216771511160194219694798, 9.568240642331605961562944213372