L(s) = 1 | + (0.707 + 0.707i)2-s + (0.292 + 1.70i)3-s + 1.00i·4-s + (−2.12 + 0.707i)5-s + (−0.999 + 1.41i)6-s + (−0.707 + 0.707i)8-s + (−2.82 + i)9-s + (−2 − 0.999i)10-s + 4.24i·11-s + (−1.70 + 0.292i)12-s + (−3 − 3i)13-s + (−1.82 − 3.41i)15-s − 1.00·16-s + (1.41 + 1.41i)17-s + (−2.70 − 1.29i)18-s − 4i·19-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.169 + 0.985i)3-s + 0.500i·4-s + (−0.948 + 0.316i)5-s + (−0.408 + 0.577i)6-s + (−0.250 + 0.250i)8-s + (−0.942 + 0.333i)9-s + (−0.632 − 0.316i)10-s + 1.27i·11-s + (−0.492 + 0.0845i)12-s + (−0.832 − 0.832i)13-s + (−0.472 − 0.881i)15-s − 0.250·16-s + (0.342 + 0.342i)17-s + (−0.638 − 0.304i)18-s − 0.917i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.337168 - 0.891041i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.337168 - 0.891041i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.292 - 1.70i)T \) |
| 5 | \( 1 + (2.12 - 0.707i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 23iT^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-8 - 8i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.24 + 4.24i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.24 - 4.24i)T - 53iT^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + (1 - i)T - 67iT^{2} \) |
| 71 | \( 1 + 1.41iT - 71T^{2} \) |
| 73 | \( 1 + (-2 - 2i)T + 73iT^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 + (-12.7 + 12.7i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + (-3 + 3i)T - 97iT^{2} \) |
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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
−10.56525388434946245156269644052, −9.755798799219257423638855381359, −8.894781933395100476992622776885, −7.80824044609798389571681304785, −7.41384087811808677296044834966, −6.24219707712050805445023914349, −4.98184243954086951435181798117, −4.56650918508809601958237593532, −3.50464013593630016629411799984, −2.62623386413265361043416781078,
0.35845225930331716462255307627, 1.78449513312604381026496152832, 3.10199337933578825288728933139, 3.89076450321693730876667080315, 5.15497109352716466242633866706, 6.02703724300821650857947394985, 7.08232546690759435107189515270, 7.79739742810154540762501862706, 8.675894411162824492333883841831, 9.407574252983029967659291355764