L(s) = 1 | − 0.339·3-s + 5-s − 3.88·7-s − 2.88·9-s − 1.54·11-s − 0.339·15-s + 2.86·19-s + 1.32·21-s − 5.42·23-s + 25-s + 2·27-s − 5.20·29-s + 6.22·31-s + 0.524·33-s − 3.88·35-s + 8.56·37-s − 9.08·41-s − 0.980·43-s − 2.88·45-s + 6.52·47-s + 8.08·49-s + 6.44·53-s − 1.54·55-s − 0.973·57-s − 4.45·59-s + 9.65·61-s + 11.2·63-s + ⋯ |
L(s) = 1 | − 0.196·3-s + 0.447·5-s − 1.46·7-s − 0.961·9-s − 0.465·11-s − 0.0877·15-s + 0.657·19-s + 0.288·21-s − 1.13·23-s + 0.200·25-s + 0.384·27-s − 0.966·29-s + 1.11·31-s + 0.0913·33-s − 0.656·35-s + 1.40·37-s − 1.41·41-s − 0.149·43-s − 0.429·45-s + 0.951·47-s + 1.15·49-s + 0.885·53-s − 0.208·55-s − 0.128·57-s − 0.580·59-s + 1.23·61-s + 1.41·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.033084759\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.033084759\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 0.339T + 3T^{2} \) |
| 7 | \( 1 + 3.88T + 7T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2.86T + 19T^{2} \) |
| 23 | \( 1 + 5.42T + 23T^{2} \) |
| 29 | \( 1 + 5.20T + 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 - 8.56T + 37T^{2} \) |
| 41 | \( 1 + 9.08T + 41T^{2} \) |
| 43 | \( 1 + 0.980T + 43T^{2} \) |
| 47 | \( 1 - 6.52T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 4.45T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 - 6.97T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 8.56T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.701597757488721226778222349651, −7.87730141612166927081359575597, −7.02625911624779116482463573803, −6.12798213734268256308011443653, −5.86227129635290645490742037074, −4.94127986577809703582793239276, −3.75844589923514895353752836833, −3.00471564472374930530643431648, −2.23195073069618392004546578871, −0.57715431274791175218619967920,
0.57715431274791175218619967920, 2.23195073069618392004546578871, 3.00471564472374930530643431648, 3.75844589923514895353752836833, 4.94127986577809703582793239276, 5.86227129635290645490742037074, 6.12798213734268256308011443653, 7.02625911624779116482463573803, 7.87730141612166927081359575597, 8.701597757488721226778222349651