L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s − 13-s + 14-s + 15-s + 16-s − 17-s − 18-s − 20-s + 21-s − 4·22-s + 4·23-s + 24-s + 25-s + 26-s − 27-s − 28-s − 2·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.223·20-s + 0.218·21-s − 0.852·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.13050828214899, −14.48178116745745, −13.88681286334037, −13.17449067258471, −12.62130962760831, −12.22583380841507, −11.67061386033276, −11.14195657521047, −10.88622169524180, −10.14776386308008, −9.561951165631954, −9.126139134091009, −8.759760136050621, −7.857367149623507, −7.482187029928672, −6.819746715220392, −6.450608292149400, −5.877806826586861, −5.115853696273506, −4.480723120822235, −3.789901812896800, −3.242247325338158, −2.372499530043682, −1.537120638554110, −0.8471306166087038, 0,
0.8471306166087038, 1.537120638554110, 2.372499530043682, 3.242247325338158, 3.789901812896800, 4.480723120822235, 5.115853696273506, 5.877806826586861, 6.450608292149400, 6.819746715220392, 7.482187029928672, 7.857367149623507, 8.759760136050621, 9.126139134091009, 9.561951165631954, 10.14776386308008, 10.88622169524180, 11.14195657521047, 11.67061386033276, 12.22583380841507, 12.62130962760831, 13.17449067258471, 13.88681286334037, 14.48178116745745, 15.13050828214899