L(s) = 1 | − 1.51·2-s + 3-s + 0.300·4-s − 2.33·5-s − 1.51·6-s + 2.57·8-s + 9-s + 3.53·10-s − 11-s + 0.300·12-s − 1.53·13-s − 2.33·15-s − 4.51·16-s − 0.116·17-s − 1.51·18-s − 3.61·19-s − 0.699·20-s + 1.51·22-s + 7.11·23-s + 2.57·24-s + 0.443·25-s + 2.33·26-s + 27-s + 5.05·29-s + 3.53·30-s − 4.37·31-s + 1.68·32-s + ⋯ |
L(s) = 1 | − 1.07·2-s + 0.577·3-s + 0.150·4-s − 1.04·5-s − 0.619·6-s + 0.911·8-s + 0.333·9-s + 1.11·10-s − 0.301·11-s + 0.0866·12-s − 0.426·13-s − 0.602·15-s − 1.12·16-s − 0.0282·17-s − 0.357·18-s − 0.828·19-s − 0.156·20-s + 0.323·22-s + 1.48·23-s + 0.526·24-s + 0.0887·25-s + 0.457·26-s + 0.192·27-s + 0.938·29-s + 0.646·30-s − 0.786·31-s + 0.297·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7303464829\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7303464829\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 1.51T + 2T^{2} \) |
| 5 | \( 1 + 2.33T + 5T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 17 | \( 1 + 0.116T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 31 | \( 1 + 4.37T + 31T^{2} \) |
| 37 | \( 1 + 0.300T + 37T^{2} \) |
| 41 | \( 1 - 8.20T + 41T^{2} \) |
| 43 | \( 1 + 4.18T + 43T^{2} \) |
| 47 | \( 1 - 2.30T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 4.95T + 59T^{2} \) |
| 61 | \( 1 + 4.57T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 4.25T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−9.241225749732460311267873282904, −8.585243774681209847408850813950, −7.949364120219675250653902466485, −7.40207017127046636224488460522, −6.60119190592521119773847925645, −5.04263203522773671540035699610, −4.32732318053608193707303076774, −3.33544612078205431019175748469, −2.13211251490002359069600695986, −0.67727340813316845915424811099,
0.67727340813316845915424811099, 2.13211251490002359069600695986, 3.33544612078205431019175748469, 4.32732318053608193707303076774, 5.04263203522773671540035699610, 6.60119190592521119773847925645, 7.40207017127046636224488460522, 7.949364120219675250653902466485, 8.585243774681209847408850813950, 9.241225749732460311267873282904