L(s) = 1 | − 2.12·2-s + 3-s + 2.52·4-s − 0.0337·5-s − 2.12·6-s + 4.05·7-s − 1.11·8-s + 9-s + 0.0718·10-s + 1.94·11-s + 2.52·12-s + 0.842·13-s − 8.61·14-s − 0.0337·15-s − 2.68·16-s + 17-s − 2.12·18-s − 4.07·19-s − 0.0852·20-s + 4.05·21-s − 4.13·22-s + 1.71·23-s − 1.11·24-s − 4.99·25-s − 1.79·26-s + 27-s + 10.2·28-s + ⋯ |
L(s) = 1 | − 1.50·2-s + 0.577·3-s + 1.26·4-s − 0.0151·5-s − 0.868·6-s + 1.53·7-s − 0.393·8-s + 0.333·9-s + 0.0227·10-s + 0.586·11-s + 0.728·12-s + 0.233·13-s − 2.30·14-s − 0.00872·15-s − 0.670·16-s + 0.242·17-s − 0.501·18-s − 0.934·19-s − 0.0190·20-s + 0.884·21-s − 0.882·22-s + 0.357·23-s − 0.227·24-s − 0.999·25-s − 0.351·26-s + 0.192·27-s + 1.93·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.527243855\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.527243855\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 5 | \( 1 + 0.0337T + 5T^{2} \) |
| 7 | \( 1 - 4.05T + 7T^{2} \) |
| 11 | \( 1 - 1.94T + 11T^{2} \) |
| 13 | \( 1 - 0.842T + 13T^{2} \) |
| 19 | \( 1 + 4.07T + 19T^{2} \) |
| 23 | \( 1 - 1.71T + 23T^{2} \) |
| 29 | \( 1 + 0.704T + 29T^{2} \) |
| 31 | \( 1 - 4.84T + 31T^{2} \) |
| 37 | \( 1 + 2.12T + 37T^{2} \) |
| 41 | \( 1 - 4.65T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 - 7.98T + 53T^{2} \) |
| 59 | \( 1 - 6.03T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 8.01T + 67T^{2} \) |
| 71 | \( 1 - 7.58T + 71T^{2} \) |
| 73 | \( 1 - 6.28T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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
−8.042989783248743666076334839460, −7.50904292112486168146303124907, −6.78947228661250754561802076185, −6.00973115515734270778598800906, −4.86659811184275000299691637446, −4.35167944073722224966568191480, −3.35190862925396062787478482777, −2.09327809135498048077649976022, −1.73980566692172040245617705411, −0.78338275165797644751336825796,
0.78338275165797644751336825796, 1.73980566692172040245617705411, 2.09327809135498048077649976022, 3.35190862925396062787478482777, 4.35167944073722224966568191480, 4.86659811184275000299691637446, 6.00973115515734270778598800906, 6.78947228661250754561802076185, 7.50904292112486168146303124907, 8.042989783248743666076334839460