L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 3.69·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s − 7.38·10-s − 2.09·11-s + 12·12-s + 92.2·13-s + 14·14-s + 11.0·15-s + 16·16-s + 38.7·17-s − 18·18-s + 26.2·19-s + 14.7·20-s − 21·21-s + 4.19·22-s + 23·23-s − 24·24-s − 111.·25-s − 184.·26-s + 27·27-s − 28·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.330·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.233·10-s − 0.0574·11-s + 0.288·12-s + 1.96·13-s + 0.267·14-s + 0.190·15-s + 0.250·16-s + 0.552·17-s − 0.235·18-s + 0.316·19-s + 0.165·20-s − 0.218·21-s + 0.0406·22-s + 0.208·23-s − 0.204·24-s − 0.890·25-s − 1.39·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.155493017\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.155493017\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
good | 5 | \( 1 - 3.69T + 125T^{2} \) |
| 11 | \( 1 + 2.09T + 1.33e3T^{2} \) |
| 13 | \( 1 - 92.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 38.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 106.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 75.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 307.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 187.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 64.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 406.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 127.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 506.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 211.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 864.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 981.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 459.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 55.0T + 5.71e5T^{2} \) |
| 89 | \( 1 - 397.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.51e3T + 9.12e5T^{2} \) |
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
−9.520733997900988327259976487707, −8.854234149537369097087208675921, −8.117067035929967962899917391371, −7.30002029754296733565561563527, −6.26571500705844299726897347524, −5.60282655718389064236261792035, −3.96846049993779665008525317149, −3.19004909101504848242828165149, −1.93532581979412381168452513153, −0.889262924272863076177702176919,
0.889262924272863076177702176919, 1.93532581979412381168452513153, 3.19004909101504848242828165149, 3.96846049993779665008525317149, 5.60282655718389064236261792035, 6.26571500705844299726897347524, 7.30002029754296733565561563527, 8.117067035929967962899917391371, 8.854234149537369097087208675921, 9.520733997900988327259976487707