L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (−1.35 − 1.77i)5-s − 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−2.21 − 0.300i)10-s + 2.44·11-s + (−0.707 − 0.707i)12-s + (−1.28 + 1.28i)13-s + (−2.21 − 0.300i)15-s − 1.00·16-s + (−5.33 − 5.33i)17-s + (−0.707 − 0.707i)18-s + 0.690·19-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s + (−0.605 − 0.795i)5-s − 0.408i·6-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + (−0.700 − 0.0951i)10-s + 0.738·11-s + (−0.204 − 0.204i)12-s + (−0.356 + 0.356i)13-s + (−0.572 − 0.0776i)15-s − 0.250·16-s + (−1.29 − 1.29i)17-s + (−0.166 − 0.166i)18-s + 0.158·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.582712065\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.582712065\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.35 + 1.77i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 2.44T + 11T^{2} \) |
| 13 | \( 1 + (1.28 - 1.28i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.33 + 5.33i)T + 17iT^{2} \) |
| 19 | \( 1 - 0.690T + 19T^{2} \) |
| 23 | \( 1 + (5.65 + 5.65i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.08iT - 29T^{2} \) |
| 31 | \( 1 + 1.26iT - 31T^{2} \) |
| 37 | \( 1 + (-7.16 + 7.16i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.15iT - 41T^{2} \) |
| 43 | \( 1 + (-0.893 - 0.893i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.95 + 4.95i)T + 47iT^{2} \) |
| 53 | \( 1 + (-6.74 - 6.74i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.06T + 59T^{2} \) |
| 61 | \( 1 + 1.09iT - 61T^{2} \) |
| 67 | \( 1 + (7.41 - 7.41i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + (4.58 - 4.58i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.31iT - 79T^{2} \) |
| 83 | \( 1 + (7.39 - 7.39i)T - 83iT^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + (4.74 + 4.74i)T + 97iT^{2} \) |
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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
−9.086716527111734391928314982464, −8.484467877181134093921278751022, −7.40383446944636209432556231495, −6.73780015308313327697835919483, −5.68131169542584440055949550876, −4.51478698264055296427411559578, −4.14611193989415403354547893269, −2.86252006076722924345385979129, −1.84639241995694315351163587954, −0.48313166395244583956521225150,
2.09088626014142770332447250235, 3.26218580347581250755440996289, 3.98651735647156781734873001572, 4.66799299042708247503010079628, 6.05604952277614070224036837002, 6.50293337108311467802804108252, 7.64647399749549330549083263862, 8.047291549888998601987502086896, 9.012827624523748460550225714619, 9.896091673520987552511361969127