L(s) = 1 | − 3.26·3-s + 1.15·5-s + 1.19·7-s + 7.65·9-s − 4.40·11-s − 5.45·13-s − 3.77·15-s + 1.75·17-s + 0.615·19-s − 3.89·21-s − 3.56·23-s − 3.66·25-s − 15.1·27-s + 5.04·29-s − 3.23·31-s + 14.3·33-s + 1.37·35-s − 8.64·37-s + 17.7·39-s − 6.82·41-s − 4.42·43-s + 8.84·45-s + 4.24·47-s − 5.57·49-s − 5.72·51-s + 9.55·53-s − 5.08·55-s + ⋯ |
L(s) = 1 | − 1.88·3-s + 0.516·5-s + 0.450·7-s + 2.55·9-s − 1.32·11-s − 1.51·13-s − 0.973·15-s + 0.425·17-s + 0.141·19-s − 0.849·21-s − 0.743·23-s − 0.732·25-s − 2.92·27-s + 0.936·29-s − 0.581·31-s + 2.50·33-s + 0.232·35-s − 1.42·37-s + 2.84·39-s − 1.06·41-s − 0.674·43-s + 1.31·45-s + 0.619·47-s − 0.796·49-s − 0.802·51-s + 1.31·53-s − 0.685·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5014760444\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5014760444\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + 3.26T + 3T^{2} \) |
| 5 | \( 1 - 1.15T + 5T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 4.40T + 11T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 19 | \( 1 - 0.615T + 19T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 + 8.64T + 37T^{2} \) |
| 41 | \( 1 + 6.82T + 41T^{2} \) |
| 43 | \( 1 + 4.42T + 43T^{2} \) |
| 47 | \( 1 - 4.24T + 47T^{2} \) |
| 53 | \( 1 - 9.55T + 53T^{2} \) |
| 59 | \( 1 - 7.62T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 1.28T + 67T^{2} \) |
| 71 | \( 1 + 2.48T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 7.25T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 + 5.02T + 97T^{2} \) |
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
−7.59967006887671663419192613887, −7.05877861068666267262463166997, −6.32858266117070448223225203910, −5.56505507845625064811191511606, −5.10226983128418500381268018950, −4.81872633626071020469465194036, −3.71199474126153180121136933027, −2.39108260079248442113981277888, −1.65300945015597719212050608597, −0.38112023836640495972036648814,
0.38112023836640495972036648814, 1.65300945015597719212050608597, 2.39108260079248442113981277888, 3.71199474126153180121136933027, 4.81872633626071020469465194036, 5.10226983128418500381268018950, 5.56505507845625064811191511606, 6.32858266117070448223225203910, 7.05877861068666267262463166997, 7.59967006887671663419192613887