L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 3·7-s + 8-s + 9-s − 10-s + 5·11-s − 12-s − 13-s − 3·14-s + 15-s + 16-s − 6·17-s + 18-s − 5·19-s − 20-s + 3·21-s + 5·22-s − 5·23-s − 24-s + 25-s − 26-s − 27-s − 3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s − 0.288·12-s − 0.277·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 1.14·19-s − 0.223·20-s + 0.654·21-s + 1.06·22-s − 1.04·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.60389951929977, −12.31402546784576, −12.00163942206916, −11.31002718182116, −10.98213209972844, −10.79141890763152, −9.992518292190100, −9.569617094092075, −9.080680748038473, −8.837459839843277, −8.012778758103292, −7.515325718123052, −6.948937910216027, −6.626275174440924, −6.183590438216659, −5.974502853418315, −5.240386786340540, −4.581302448327290, −4.152013630123130, −3.842125061927690, −3.427609663621943, −2.623828119622792, −2.005188165058882, −1.604135755322180, −0.5492877811854213, 0,
0.5492877811854213, 1.604135755322180, 2.005188165058882, 2.623828119622792, 3.427609663621943, 3.842125061927690, 4.152013630123130, 4.581302448327290, 5.240386786340540, 5.974502853418315, 6.183590438216659, 6.626275174440924, 6.948937910216027, 7.515325718123052, 8.012778758103292, 8.837459839843277, 9.080680748038473, 9.569617094092075, 9.992518292190100, 10.79141890763152, 10.98213209972844, 11.31002718182116, 12.00163942206916, 12.31402546784576, 12.60389951929977