L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 2·11-s − 13-s + 2·14-s + 16-s − 7·19-s − 20-s − 2·22-s + 2·23-s + 25-s + 26-s − 2·28-s + 2·29-s − 10·31-s − 32-s + 2·35-s + 3·37-s + 7·38-s + 40-s + 5·41-s − 6·43-s + 2·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.603·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.60·19-s − 0.223·20-s − 0.426·22-s + 0.417·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s + 0.371·29-s − 1.79·31-s − 0.176·32-s + 0.338·35-s + 0.493·37-s + 1.13·38-s + 0.158·40-s + 0.780·41-s − 0.914·43-s + 0.301·44-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
\(L(1)\) |
\(\approx\) |
\(0.1965959109\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1965959109\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−12.45213672367882, −12.23393008110098, −11.55097226747068, −11.16197826398263, −10.66439164547362, −10.43036957804281, −9.746183513402299, −9.280660374679356, −9.005632096935162, −8.598122562841965, −7.888424912465032, −7.632010454672567, −7.013609549822022, −6.591163746228303, −6.219909683482048, −5.741550416092103, −5.000521156263931, −4.410473677326160, −3.991883514892619, −3.328398492008749, −2.960902581065359, −2.211477028273097, −1.720261903319122, −1.005931204882889, −0.1439682889361424,
0.1439682889361424, 1.005931204882889, 1.720261903319122, 2.211477028273097, 2.960902581065359, 3.328398492008749, 3.991883514892619, 4.410473677326160, 5.000521156263931, 5.741550416092103, 6.219909683482048, 6.591163746228303, 7.013609549822022, 7.632010454672567, 7.888424912465032, 8.598122562841965, 9.005632096935162, 9.280660374679356, 9.746183513402299, 10.43036957804281, 10.66439164547362, 11.16197826398263, 11.55097226747068, 12.23393008110098, 12.45213672367882