L(s) = 1 | − 2-s + 3.10·3-s + 4-s + 5-s − 3.10·6-s − 7-s − 8-s + 6.62·9-s − 10-s + 3.10·12-s + 14-s + 3.10·15-s + 16-s − 5.62·17-s − 6.62·18-s + 4.72·19-s + 20-s − 3.10·21-s + 4.52·23-s − 3.10·24-s + 25-s + 11.2·27-s − 28-s + 2.52·29-s − 3.10·30-s − 3.04·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.79·3-s + 0.5·4-s + 0.447·5-s − 1.26·6-s − 0.377·7-s − 0.353·8-s + 2.20·9-s − 0.316·10-s + 0.895·12-s + 0.267·14-s + 0.801·15-s + 0.250·16-s − 1.36·17-s − 1.56·18-s + 1.08·19-s + 0.223·20-s − 0.677·21-s + 0.943·23-s − 0.633·24-s + 0.200·25-s + 2.16·27-s − 0.188·28-s + 0.468·29-s − 0.566·30-s − 0.547·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.395574631\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.395574631\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5.62T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 + 3.04T + 31T^{2} \) |
| 37 | \( 1 - 4.52T + 37T^{2} \) |
| 41 | \( 1 - 7.30T + 41T^{2} \) |
| 43 | \( 1 + 0.578T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 5.57T + 53T^{2} \) |
| 59 | \( 1 + 7.62T + 59T^{2} \) |
| 61 | \( 1 - 9.83T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 8.41T + 71T^{2} \) |
| 73 | \( 1 - 4.57T + 73T^{2} \) |
| 79 | \( 1 + 9.77T + 79T^{2} \) |
| 83 | \( 1 - 7.25T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.72834131677281688576672337815, −7.43571806810493118150941218626, −6.70964770507679914891098966921, −5.95558643533625662167774421323, −4.82830616182995868583518684837, −4.03490312361028819506401755127, −3.12756591285512362338790717020, −2.63743298800753939922094079438, −1.91879505144826918516533917913, −0.941826935418002484495265503402,
0.941826935418002484495265503402, 1.91879505144826918516533917913, 2.63743298800753939922094079438, 3.12756591285512362338790717020, 4.03490312361028819506401755127, 4.82830616182995868583518684837, 5.95558643533625662167774421323, 6.70964770507679914891098966921, 7.43571806810493118150941218626, 7.72834131677281688576672337815