L(s) = 1 | − 2-s − 2.53·3-s + 4-s − 5-s + 2.53·6-s + 7-s − 8-s + 3.44·9-s + 10-s − 2.53·12-s + 4.11·13-s − 14-s + 2.53·15-s + 16-s + 6.84·17-s − 3.44·18-s + 6.62·19-s − 20-s − 2.53·21-s − 3.13·23-s + 2.53·24-s + 25-s − 4.11·26-s − 1.13·27-s + 28-s − 5.49·29-s − 2.53·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.46·3-s + 0.5·4-s − 0.447·5-s + 1.03·6-s + 0.377·7-s − 0.353·8-s + 1.14·9-s + 0.316·10-s − 0.733·12-s + 1.14·13-s − 0.267·14-s + 0.655·15-s + 0.250·16-s + 1.66·17-s − 0.812·18-s + 1.51·19-s − 0.223·20-s − 0.554·21-s − 0.654·23-s + 0.518·24-s + 0.200·25-s − 0.806·26-s − 0.219·27-s + 0.188·28-s − 1.02·29-s − 0.463·30-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.53T + 3T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 - 6.84T + 17T^{2} \) |
| 19 | \( 1 - 6.62T + 19T^{2} \) |
| 23 | \( 1 + 3.13T + 23T^{2} \) |
| 29 | \( 1 + 5.49T + 29T^{2} \) |
| 31 | \( 1 + 5.57T + 31T^{2} \) |
| 37 | \( 1 + 7.78T + 37T^{2} \) |
| 41 | \( 1 + 2.90T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 + 1.49T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 7.42T + 59T^{2} \) |
| 61 | \( 1 - 2.28T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 0.0608T + 71T^{2} \) |
| 73 | \( 1 - 3.91T + 73T^{2} \) |
| 79 | \( 1 - 4.23T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 6.54T + 89T^{2} \) |
| 97 | \( 1 + 0.691T + 97T^{2} \) |
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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
−7.47340540639653516283522190656, −6.80955284778155696405651636229, −5.98881732168822089029876998051, −5.47808909109441333802160870781, −4.98644183221382929512777296018, −3.74190670794476643842959848506, −3.28068264940698269598165466902, −1.65833667611238168708439514274, −1.07761383722983084508320568114, 0,
1.07761383722983084508320568114, 1.65833667611238168708439514274, 3.28068264940698269598165466902, 3.74190670794476643842959848506, 4.98644183221382929512777296018, 5.47808909109441333802160870781, 5.98881732168822089029876998051, 6.80955284778155696405651636229, 7.47340540639653516283522190656