L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s + 5·11-s − 12-s + 13-s − 15-s + 16-s + 3·17-s + 18-s + 19-s + 20-s + 5·22-s + 3·23-s − 24-s − 4·25-s + 26-s − 27-s + 9·29-s − 30-s − 4·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 1.06·22-s + 0.625·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.192·27-s + 1.67·29-s − 0.182·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.239143392\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.239143392\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.570270957245060272556771742531, −7.46707625018519375379749670353, −6.78495406876246844747627680662, −6.21061604242769480320897121006, −5.52271654641630540580854974374, −4.79825735516132432727394797382, −3.89643097901644695246306643209, −3.22453570458517048163514654338, −1.90398148609377823300526796355, −1.05036223063446686564800493982,
1.05036223063446686564800493982, 1.90398148609377823300526796355, 3.22453570458517048163514654338, 3.89643097901644695246306643209, 4.79825735516132432727394797382, 5.52271654641630540580854974374, 6.21061604242769480320897121006, 6.78495406876246844747627680662, 7.46707625018519375379749670353, 8.570270957245060272556771742531