L(s) = 1 | + 2-s + 3-s − 4-s + 6-s − 3·8-s + 9-s + 4·11-s − 12-s − 13-s − 16-s − 17-s + 18-s − 4·19-s + 4·22-s − 3·24-s − 26-s + 27-s − 2·29-s − 8·31-s + 5·32-s + 4·33-s − 34-s − 36-s + 2·37-s − 4·38-s − 39-s + 2·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 0.277·13-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s + 0.852·22-s − 0.612·24-s − 0.196·26-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.883·32-s + 0.696·33-s − 0.171·34-s − 1/6·36-s + 0.328·37-s − 0.648·38-s − 0.160·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.866428151\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.866428151\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.81682062252297, −14.95716563077976, −14.70283991391010, −14.40240275208779, −13.76225058434183, −13.08572695020469, −12.79799775055427, −12.24568740018540, −11.47885920198090, −11.08984210241083, −10.08963420646601, −9.608770183938832, −9.045150150500018, −8.619905510487857, −7.996532601302048, −7.110097016500909, −6.590163014656709, −5.909660767803284, −5.200873306065482, −4.500909017198959, −3.852993691072989, −3.546795691462327, −2.523751063203468, −1.802609320415069, −0.6249709372056015,
0.6249709372056015, 1.802609320415069, 2.523751063203468, 3.546795691462327, 3.852993691072989, 4.500909017198959, 5.200873306065482, 5.909660767803284, 6.590163014656709, 7.110097016500909, 7.996532601302048, 8.619905510487857, 9.045150150500018, 9.608770183938832, 10.08963420646601, 11.08984210241083, 11.47885920198090, 12.24568740018540, 12.79799775055427, 13.08572695020469, 13.76225058434183, 14.40240275208779, 14.70283991391010, 14.95716563077976, 15.81682062252297