L(s) = 1 | + (0.5 − 0.866i)5-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (1.5 − 0.866i)17-s + (−0.499 − 0.866i)25-s + (0.5 − 0.866i)29-s + (−1.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + 0.999·45-s + (0.5 − 0.866i)49-s + 1.73i·53-s + (−0.5 − 0.866i)61-s + (0.499 + 0.866i)65-s + 1.73i·73-s + (−0.499 + 0.866i)81-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)13-s + (1.5 − 0.866i)17-s + (−0.499 − 0.866i)25-s + (0.5 − 0.866i)29-s + (−1.5 − 0.866i)37-s + (−0.5 + 0.866i)41-s + 0.999·45-s + (0.5 − 0.866i)49-s + 1.73i·53-s + (−0.5 − 0.866i)61-s + (0.499 + 0.866i)65-s + 1.73i·73-s + (−0.499 + 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.159229963\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.159229963\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.73iT - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - 1.73iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.891168385713142400102776488190, −9.449384519932893585450131467505, −8.429520864978276511742761606277, −7.65058065203721022523195252301, −6.80830915424578625969912651996, −5.57753595783975263062275306693, −4.98019974557407357043237881549, −4.07490190624807278077070041448, −2.55502754796435606772442979893, −1.42418175482827334192559199993,
1.53220557949513475894902935665, 3.02912494137979756079521566552, 3.66518828928059960971808040887, 5.13766913926084535926550830737, 5.95478592302539476574271026437, 6.81474563827652541560613109977, 7.52772615669312912319781835329, 8.510880341095104615630608931758, 9.542713035633365644607300129399, 10.29419586511611078018766379543