L(s) = 1 | − 1.23·2-s + 0.363·3-s − 0.477·4-s − 0.449·6-s − 2.58·7-s + 3.05·8-s − 2.86·9-s − 0.173·12-s − 2.75·13-s + 3.19·14-s − 2.81·16-s + 3.85·17-s + 3.53·18-s − 0.277·19-s − 0.941·21-s − 8.40·23-s + 1.11·24-s + 3.40·26-s − 2.13·27-s + 1.23·28-s + 3.32·29-s + 0.564·31-s − 2.63·32-s − 4.75·34-s + 1.36·36-s − 0.522·37-s + 0.342·38-s + ⋯ |
L(s) = 1 | − 0.872·2-s + 0.210·3-s − 0.238·4-s − 0.183·6-s − 0.977·7-s + 1.08·8-s − 0.955·9-s − 0.0501·12-s − 0.765·13-s + 0.852·14-s − 0.704·16-s + 0.934·17-s + 0.834·18-s − 0.0636·19-s − 0.205·21-s − 1.75·23-s + 0.227·24-s + 0.667·26-s − 0.411·27-s + 0.233·28-s + 0.617·29-s + 0.101·31-s − 0.466·32-s − 0.815·34-s + 0.228·36-s − 0.0859·37-s + 0.0555·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4930945842\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4930945842\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.23T + 2T^{2} \) |
| 3 | \( 1 - 0.363T + 3T^{2} \) |
| 7 | \( 1 + 2.58T + 7T^{2} \) |
| 13 | \( 1 + 2.75T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 + 0.277T + 19T^{2} \) |
| 23 | \( 1 + 8.40T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 - 0.564T + 31T^{2} \) |
| 37 | \( 1 + 0.522T + 37T^{2} \) |
| 41 | \( 1 + 5.11T + 41T^{2} \) |
| 43 | \( 1 - 2.54T + 43T^{2} \) |
| 47 | \( 1 + 4.92T + 47T^{2} \) |
| 53 | \( 1 + 8.72T + 53T^{2} \) |
| 59 | \( 1 - 7.50T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 3.20T + 67T^{2} \) |
| 71 | \( 1 + 8.40T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 - 9.70T + 79T^{2} \) |
| 83 | \( 1 - 3.29T + 83T^{2} \) |
| 89 | \( 1 - 2.48T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−8.665898859786933237272351881601, −8.150044606420095317052603048432, −7.49504190725843631216566402060, −6.56013727047276331620415298725, −5.75666102769790132850260739050, −4.90184307159470702278358140547, −3.84806639144419988667357572253, −3.02534492552064555196025035788, −1.95828588352826110320918324804, −0.46590008332972863942747306324,
0.46590008332972863942747306324, 1.95828588352826110320918324804, 3.02534492552064555196025035788, 3.84806639144419988667357572253, 4.90184307159470702278358140547, 5.75666102769790132850260739050, 6.56013727047276331620415298725, 7.49504190725843631216566402060, 8.150044606420095317052603048432, 8.665898859786933237272351881601