L(s) = 1 | + i·2-s − 4-s + 5-s − i·8-s + i·10-s + 5.39i·11-s − 2.51i·13-s + 16-s − 4.49·17-s − 2.86i·19-s − 20-s − 5.39·22-s + 0.267i·23-s + 25-s + 2.51·26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.447·5-s − 0.353i·8-s + 0.316i·10-s + 1.62i·11-s − 0.698i·13-s + 0.250·16-s − 1.09·17-s − 0.656i·19-s − 0.223·20-s − 1.15·22-s + 0.0558i·23-s + 0.200·25-s + 0.493·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6849827146\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6849827146\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5.39iT - 11T^{2} \) |
| 13 | \( 1 + 2.51iT - 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 + 2.86iT - 19T^{2} \) |
| 23 | \( 1 - 0.267iT - 23T^{2} \) |
| 29 | \( 1 + 8.89iT - 29T^{2} \) |
| 31 | \( 1 + 4.82iT - 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 - 0.760T + 41T^{2} \) |
| 43 | \( 1 + 5.86T + 43T^{2} \) |
| 47 | \( 1 + 7.99T + 47T^{2} \) |
| 53 | \( 1 - 8.39iT - 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2.62iT - 61T^{2} \) |
| 67 | \( 1 - 9.82T + 67T^{2} \) |
| 71 | \( 1 + 4.76iT - 71T^{2} \) |
| 73 | \( 1 + 11.6iT - 73T^{2} \) |
| 79 | \( 1 - 8.59T + 79T^{2} \) |
| 83 | \( 1 - 9.45T + 83T^{2} \) |
| 89 | \( 1 - 7.97T + 89T^{2} \) |
| 97 | \( 1 + 6.16iT - 97T^{2} \) |
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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
−7.931860598616670691670900405356, −7.58113798575775953210876871326, −6.58168116905091621803984300511, −6.28565194158479058833206554246, −5.11226847367260277006131425519, −4.72979298856271288963838555487, −3.84110977620220671912999841911, −2.59521147542846716598627070179, −1.78700580336920732157079638011, −0.18813774500034311289542452344,
1.24232614829045070628351906316, 2.08299871999283245235007685728, 3.19399256958539664093000864186, 3.68387675236478587285646234069, 4.83531909201346583510749258380, 5.40173246468487851404374652398, 6.39179948618767201858329576469, 6.84266314063041287575063095153, 8.095796774785902736092903147699, 8.706549375349547149342160542163