L(s) = 1 | + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (1.08 + 1.63i)5-s + (0.382 − 0.923i)8-s + (0.923 + 0.382i)9-s + (1.63 + 1.08i)10-s + (−1 + i)13-s − i·16-s + (−0.382 − 0.923i)17-s + 18-s + (1.92 + 0.382i)20-s + (−1.08 + 2.63i)25-s + (−0.541 + 1.30i)26-s + (−1.08 − 1.63i)29-s + (−0.382 − 0.923i)32-s + ⋯ |
L(s) = 1 | + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (1.08 + 1.63i)5-s + (0.382 − 0.923i)8-s + (0.923 + 0.382i)9-s + (1.63 + 1.08i)10-s + (−1 + i)13-s − i·16-s + (−0.382 − 0.923i)17-s + 18-s + (1.92 + 0.382i)20-s + (−1.08 + 2.63i)25-s + (−0.541 + 1.30i)26-s + (−1.08 − 1.63i)29-s + (−0.382 − 0.923i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.732340466\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.732340466\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 + 0.382i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.382 + 0.923i)T \) |
good | 3 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 5 | \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \) |
| 11 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 13 | \( 1 + (1 - i)T - iT^{2} \) |
| 19 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 29 | \( 1 + (1.08 + 1.63i)T + (-0.382 + 0.923i)T^{2} \) |
| 31 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 37 | \( 1 + (0.0761 - 0.382i)T + (-0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (-0.617 + 0.923i)T + (-0.382 - 0.923i)T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (1.38 + 0.923i)T + (0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 73 | \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \) |
| 79 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \) |
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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
−9.436159429142385511348501309756, −7.48630868532192889407489944524, −7.23962112203092582071172051827, −6.49697655073959609450608583921, −5.88875164885539569733987236705, −4.94626846915300455026515513030, −4.19042157806563526349719489177, −3.16936810715799331158382917920, −2.25995878626728419718030403788, −1.91119600244485999334217766606,
1.36678845858280938826912420401, 2.17660335775675869784094409658, 3.43566043374277552351188392069, 4.48445383698681709180413472505, 4.91370637756199033564322352478, 5.68248807870316830422451670137, 6.23851949515265335598200283216, 7.24281537781091256121382202001, 7.936445552035025418717927714435, 8.769684432833278011057139629084