L(s) = 1 | − 4.69·2-s + 14.0·4-s − 4.47·5-s + 27.2·7-s − 28.5·8-s + 21.0·10-s + 5.99·11-s − 127.·14-s + 21.4·16-s − 105.·17-s − 156.·19-s − 62.9·20-s − 28.1·22-s + 175.·23-s − 104.·25-s + 382.·28-s − 204.·29-s + 31.9·31-s + 127.·32-s + 493.·34-s − 121.·35-s + 344.·37-s + 735.·38-s + 127.·40-s − 46.5·41-s − 173.·43-s + 84.3·44-s + ⋯ |
L(s) = 1 | − 1.66·2-s + 1.75·4-s − 0.400·5-s + 1.46·7-s − 1.26·8-s + 0.664·10-s + 0.164·11-s − 2.44·14-s + 0.335·16-s − 1.49·17-s − 1.88·19-s − 0.703·20-s − 0.272·22-s + 1.59·23-s − 0.839·25-s + 2.58·28-s − 1.31·29-s + 0.185·31-s + 0.703·32-s + 2.48·34-s − 0.587·35-s + 1.52·37-s + 3.13·38-s + 0.504·40-s − 0.177·41-s − 0.614·43-s + 0.288·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7502021659\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7502021659\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 4.69T + 8T^{2} \) |
| 5 | \( 1 + 4.47T + 125T^{2} \) |
| 7 | \( 1 - 27.2T + 343T^{2} \) |
| 11 | \( 1 - 5.99T + 1.33e3T^{2} \) |
| 17 | \( 1 + 105.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 156.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 175.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 31.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 344.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 46.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 173.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 265.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 137.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 58.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 211.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 436.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 150.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 565.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 286.T + 9.12e5T^{2} \) |
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
−8.852274707817656337517678891168, −8.480428109512516860191988496118, −7.75234160209829514189795603518, −7.05010108404385683044093028381, −6.21122230329908481364366169286, −4.84394670936646214543515512788, −4.10318080804921349120461086780, −2.36820139318118638757909275908, −1.73896847152893431464460164442, −0.54403998776489265633698222335,
0.54403998776489265633698222335, 1.73896847152893431464460164442, 2.36820139318118638757909275908, 4.10318080804921349120461086780, 4.84394670936646214543515512788, 6.21122230329908481364366169286, 7.05010108404385683044093028381, 7.75234160209829514189795603518, 8.480428109512516860191988496118, 8.852274707817656337517678891168