L(s) = 1 | − 2.36·2-s − 1.70·3-s + 3.59·4-s − 0.509·5-s + 4.03·6-s + 0.312·7-s − 3.77·8-s − 0.0864·9-s + 1.20·10-s + 11-s − 6.13·12-s + 3.76·13-s − 0.739·14-s + 0.870·15-s + 1.73·16-s − 17-s + 0.204·18-s − 3.37·19-s − 1.83·20-s − 0.533·21-s − 2.36·22-s + 0.611·23-s + 6.43·24-s − 4.74·25-s − 8.89·26-s + 5.26·27-s + 1.12·28-s + ⋯ |
L(s) = 1 | − 1.67·2-s − 0.985·3-s + 1.79·4-s − 0.227·5-s + 1.64·6-s + 0.118·7-s − 1.33·8-s − 0.0288·9-s + 0.381·10-s + 0.301·11-s − 1.77·12-s + 1.04·13-s − 0.197·14-s + 0.224·15-s + 0.432·16-s − 0.242·17-s + 0.0482·18-s − 0.773·19-s − 0.409·20-s − 0.116·21-s − 0.504·22-s + 0.127·23-s + 1.31·24-s − 0.948·25-s − 1.74·26-s + 1.01·27-s + 0.212·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2535060686\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2535060686\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 3 | \( 1 + 1.70T + 3T^{2} \) |
| 5 | \( 1 + 0.509T + 5T^{2} \) |
| 7 | \( 1 - 0.312T + 7T^{2} \) |
| 13 | \( 1 - 3.76T + 13T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 - 0.611T + 23T^{2} \) |
| 29 | \( 1 + 7.27T + 29T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 + 6.00T + 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 47 | \( 1 + 7.02T + 47T^{2} \) |
| 53 | \( 1 + 4.25T + 53T^{2} \) |
| 59 | \( 1 - 9.41T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 - 9.83T + 67T^{2} \) |
| 71 | \( 1 + 3.92T + 71T^{2} \) |
| 73 | \( 1 - 1.30T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 4.52T + 83T^{2} \) |
| 89 | \( 1 + 7.93T + 89T^{2} \) |
| 97 | \( 1 - 1.17T + 97T^{2} \) |
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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
−8.122271805570202366518157393448, −7.09632300625134062966295903561, −6.72133215025475371068488007254, −5.91628942045965223757163547155, −5.38608829625047922748552008368, −4.24026215577929414668550178841, −3.44611187492943470313131482772, −2.15234887382867363414814508230, −1.44850222755291737403931110970, −0.35302735527974503451004733084,
0.35302735527974503451004733084, 1.44850222755291737403931110970, 2.15234887382867363414814508230, 3.44611187492943470313131482772, 4.24026215577929414668550178841, 5.38608829625047922748552008368, 5.91628942045965223757163547155, 6.72133215025475371068488007254, 7.09632300625134062966295903561, 8.122271805570202366518157393448