L(s) = 1 | − 4.82·2-s + 8.55·3-s + 15.2·4-s − 5·5-s − 41.2·6-s − 30.1·7-s − 35.0·8-s + 46.2·9-s + 24.1·10-s + 11·11-s + 130.·12-s − 23.7·13-s + 145.·14-s − 42.7·15-s + 46.7·16-s − 18.3·17-s − 223.·18-s − 19·19-s − 76.3·20-s − 257.·21-s − 53.0·22-s + 212.·23-s − 299.·24-s + 25·25-s + 114.·26-s + 164.·27-s − 459.·28-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 1.64·3-s + 1.90·4-s − 0.447·5-s − 2.80·6-s − 1.62·7-s − 1.54·8-s + 1.71·9-s + 0.762·10-s + 0.301·11-s + 3.14·12-s − 0.507·13-s + 2.77·14-s − 0.736·15-s + 0.731·16-s − 0.261·17-s − 2.92·18-s − 0.229·19-s − 0.853·20-s − 2.68·21-s − 0.514·22-s + 1.92·23-s − 2.54·24-s + 0.200·25-s + 0.865·26-s + 1.17·27-s − 3.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
good | 2 | \( 1 + 4.82T + 8T^{2} \) |
| 3 | \( 1 - 8.55T + 27T^{2} \) |
| 7 | \( 1 + 30.1T + 343T^{2} \) |
| 13 | \( 1 + 23.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.3T + 4.91e3T^{2} \) |
| 23 | \( 1 - 212.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 237.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 230.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 331.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 449.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 38.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + 345.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 888.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 818.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 701.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 193.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.05e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 443.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 938.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.34e3T + 9.12e5T^{2} \) |
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
−9.083826557201761200836860703199, −8.618297134298029550621965916750, −7.63449698482302903343593455092, −7.09126979965881782438886815444, −6.37270748735187249592203954349, −4.36343978506758938124703677974, −3.03684430708779688394419872091, −2.75587584211636830782526541817, −1.30157788456569080038195051116, 0,
1.30157788456569080038195051116, 2.75587584211636830782526541817, 3.03684430708779688394419872091, 4.36343978506758938124703677974, 6.37270748735187249592203954349, 7.09126979965881782438886815444, 7.63449698482302903343593455092, 8.618297134298029550621965916750, 9.083826557201761200836860703199