L(s) = 1 | − 2-s + 4-s − 8-s − 6·11-s − 13-s + 16-s + 19-s + 6·22-s − 6·23-s + 26-s + 3·29-s − 8·31-s − 32-s − 37-s − 38-s − 9·41-s + 8·43-s − 6·44-s + 6·46-s − 3·47-s − 52-s − 3·53-s − 3·58-s − 6·59-s + 10·61-s + 8·62-s + 64-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s − 0.277·13-s + 1/4·16-s + 0.229·19-s + 1.27·22-s − 1.25·23-s + 0.196·26-s + 0.557·29-s − 1.43·31-s − 0.176·32-s − 0.164·37-s − 0.162·38-s − 1.40·41-s + 1.21·43-s − 0.904·44-s + 0.884·46-s − 0.437·47-s − 0.138·52-s − 0.412·53-s − 0.393·58-s − 0.781·59-s + 1.28·61-s + 1.01·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.03442646786995, −12.31747036008475, −12.10284024079808, −11.56221968786957, −10.86833732873867, −10.64146759084043, −10.25805206925063, −9.747409773338561, −9.342340008634300, −8.763775852700085, −8.235420076373582, −7.877444595780707, −7.503558090910411, −7.043522778821281, −6.410262060404334, −5.864196279595900, −5.407854604258643, −4.971185832685207, −4.374276897972867, −3.626262226968822, −3.128149646781361, −2.531604345835929, −2.055477600828272, −1.509910161983639, −0.5320631161282894, 0,
0.5320631161282894, 1.509910161983639, 2.055477600828272, 2.531604345835929, 3.128149646781361, 3.626262226968822, 4.374276897972867, 4.971185832685207, 5.407854604258643, 5.864196279595900, 6.410262060404334, 7.043522778821281, 7.503558090910411, 7.877444595780707, 8.235420076373582, 8.763775852700085, 9.342340008634300, 9.747409773338561, 10.25805206925063, 10.64146759084043, 10.86833732873867, 11.56221968786957, 12.10284024079808, 12.31747036008475, 13.03442646786995