| L(s) = 1 | + 2-s − 2.16·3-s + 4-s + 1.79·5-s − 2.16·6-s + 4.71·7-s + 8-s + 1.70·9-s + 1.79·10-s − 2.16·12-s + 6.85·13-s + 4.71·14-s − 3.88·15-s + 16-s − 0.0921·17-s + 1.70·18-s − 19-s + 1.79·20-s − 10.2·21-s + 4.91·23-s − 2.16·24-s − 1.79·25-s + 6.85·26-s + 2.80·27-s + 4.71·28-s − 3.20·29-s − 3.88·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.25·3-s + 0.5·4-s + 0.800·5-s − 0.885·6-s + 1.78·7-s + 0.353·8-s + 0.569·9-s + 0.566·10-s − 0.626·12-s + 1.90·13-s + 1.25·14-s − 1.00·15-s + 0.250·16-s − 0.0223·17-s + 0.402·18-s − 0.229·19-s + 0.400·20-s − 2.23·21-s + 1.02·23-s − 0.442·24-s − 0.358·25-s + 1.34·26-s + 0.539·27-s + 0.890·28-s − 0.595·29-s − 0.709·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.372160935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.372160935\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.16T + 3T^{2} \) |
| 5 | \( 1 - 1.79T + 5T^{2} \) |
| 7 | \( 1 - 4.71T + 7T^{2} \) |
| 13 | \( 1 - 6.85T + 13T^{2} \) |
| 17 | \( 1 + 0.0921T + 17T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 - 5.91T + 37T^{2} \) |
| 41 | \( 1 - 3.60T + 41T^{2} \) |
| 43 | \( 1 + 7.91T + 43T^{2} \) |
| 47 | \( 1 - 2.78T + 47T^{2} \) |
| 53 | \( 1 + 7.27T + 53T^{2} \) |
| 59 | \( 1 - 8.16T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 5.87T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 2.00T + 83T^{2} \) |
| 89 | \( 1 + 9.13T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.300819010683617596911355768535, −7.44666853198922347717676662602, −6.46840631492725403181486239909, −5.94421791051874683089540304536, −5.48620522994515029093582989951, −4.71410977805448181384715667757, −4.18529960809976432872077752648, −2.90332588850299123552387480864, −1.66744700966558004933409044786, −1.13388976026901610095847637915,
1.13388976026901610095847637915, 1.66744700966558004933409044786, 2.90332588850299123552387480864, 4.18529960809976432872077752648, 4.71410977805448181384715667757, 5.48620522994515029093582989951, 5.94421791051874683089540304536, 6.46840631492725403181486239909, 7.44666853198922347717676662602, 8.300819010683617596911355768535
