L(s) = 1 | + (−1 − i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (4.88 + 1.07i)5-s − 2.44·6-s + (6.40 + 6.40i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−3.80 − 5.96i)10-s − 0.735·11-s + (2.44 + 2.44i)12-s + (6.53 − 6.53i)13-s − 12.8i·14-s + (7.30 − 4.65i)15-s − 4·16-s + (−3.30 − 3.30i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (0.976 + 0.215i)5-s − 0.408·6-s + (0.915 + 0.915i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (−0.380 − 0.596i)10-s − 0.0668·11-s + (0.204 + 0.204i)12-s + (0.503 − 0.503i)13-s − 0.915i·14-s + (0.486 − 0.310i)15-s − 0.250·16-s + (−0.194 − 0.194i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.241304901\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.241304901\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 + (-4.88 - 1.07i)T \) |
| 23 | \( 1 + (3.39 - 3.39i)T \) |
good | 7 | \( 1 + (-6.40 - 6.40i)T + 49iT^{2} \) |
| 11 | \( 1 + 0.735T + 121T^{2} \) |
| 13 | \( 1 + (-6.53 + 6.53i)T - 169iT^{2} \) |
| 17 | \( 1 + (3.30 + 3.30i)T + 289iT^{2} \) |
| 19 | \( 1 + 3.16iT - 361T^{2} \) |
| 29 | \( 1 - 0.982iT - 841T^{2} \) |
| 31 | \( 1 - 29.3T + 961T^{2} \) |
| 37 | \( 1 + (-39.1 - 39.1i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 2.92T + 1.68e3T^{2} \) |
| 43 | \( 1 + (29.2 - 29.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-18.9 - 18.9i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (2.56 - 2.56i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 52.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 49.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + (28.7 + 28.7i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 15.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-22.2 + 22.2i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 - 103. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-37.5 + 37.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 21.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (85.8 + 85.8i)T + 9.40e3iT^{2} \) |
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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
−10.08219459304626535031982951174, −9.350066016104798164693861897003, −8.491813836277787378650387723800, −7.953350839533909357173223365588, −6.70330958286023593141276453204, −5.79064647963092591301571476424, −4.72570965189279743114200910245, −3.07479612089268115843930394971, −2.24829406388456445362769434101, −1.24591752026972612204473303780,
1.12613758655623886603117148432, 2.25880684233138069650385354978, 4.00974827409125889842086491835, 4.87956802181678991667282465274, 5.90684700718290665104036028040, 6.88504269942746642497453756337, 7.87312811903232845006054858604, 8.631207298250226850694261474261, 9.377320760024730384120464006632, 10.28135492914636153457802785101