L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 4·7-s + 8-s + 9-s − 10-s − 12-s − 2·13-s − 4·14-s + 15-s + 16-s + 6·17-s + 18-s − 20-s + 4·21-s − 24-s + 25-s − 2·26-s − 27-s − 4·28-s + 6·29-s + 30-s − 8·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.554·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.223·20-s + 0.872·21-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + 0.182·30-s − 1.43·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.80831784673487, −16.20603008232720, −15.71199400831191, −15.20906563516326, −14.45294156900307, −14.01142714413248, −13.19301323535288, −12.64047631318110, −12.36166002151993, −11.82383143909755, −11.13293091054548, −10.35184964811019, −9.996001673329752, −9.346634076879411, −8.520583335198805, −7.490178803901675, −7.251523322584130, −6.415022139714829, −5.924693091037587, −5.263874561607027, −4.545971881698166, −3.631928854200030, −3.268375570797653, −2.380297776596413, −1.083908573673478, 0,
1.083908573673478, 2.380297776596413, 3.268375570797653, 3.631928854200030, 4.545971881698166, 5.263874561607027, 5.924693091037587, 6.415022139714829, 7.251523322584130, 7.490178803901675, 8.520583335198805, 9.346634076879411, 9.996001673329752, 10.35184964811019, 11.13293091054548, 11.82383143909755, 12.36166002151993, 12.64047631318110, 13.19301323535288, 14.01142714413248, 14.45294156900307, 15.20906563516326, 15.71199400831191, 16.20603008232720, 16.80831784673487