L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 2·11-s − 13-s + 16-s − 2·19-s − 20-s − 2·22-s − 6·23-s + 25-s + 26-s + 2·29-s − 6·31-s − 32-s + 10·37-s + 2·38-s + 40-s + 2·41-s − 2·43-s + 2·44-s + 6·46-s + 4·47-s − 7·49-s − 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.603·11-s − 0.277·13-s + 1/4·16-s − 0.458·19-s − 0.223·20-s − 0.426·22-s − 1.25·23-s + 1/5·25-s + 0.196·26-s + 0.371·29-s − 1.07·31-s − 0.176·32-s + 1.64·37-s + 0.324·38-s + 0.158·40-s + 0.312·41-s − 0.304·43-s + 0.301·44-s + 0.884·46-s + 0.583·47-s − 49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.321647508\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.321647508\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−12.43510182584469, −12.01881478255580, −11.60266031951721, −11.29230559926769, −10.66774655476802, −10.34879214066379, −9.731362264810331, −9.459223657801772, −8.909723723917293, −8.414520469078634, −8.073795529367561, −7.507536615177800, −7.185375303570649, −6.568011993980582, −6.147284624382959, −5.706559011664705, −5.034448507183304, −4.406082162551763, −3.956789467940065, −3.516341109558107, −2.777085122068566, −2.224246775825056, −1.741932205285121, −0.9506178980969311, −0.3956919805450207,
0.3956919805450207, 0.9506178980969311, 1.741932205285121, 2.224246775825056, 2.777085122068566, 3.516341109558107, 3.956789467940065, 4.406082162551763, 5.034448507183304, 5.706559011664705, 6.147284624382959, 6.568011993980582, 7.185375303570649, 7.507536615177800, 8.073795529367561, 8.414520469078634, 8.909723723917293, 9.459223657801772, 9.731362264810331, 10.34879214066379, 10.66774655476802, 11.29230559926769, 11.60266031951721, 12.01881478255580, 12.43510182584469