L(s) = 1 | − 0.414·2-s − 1.82·4-s + 2·5-s + 1.58·8-s − 0.828·10-s − 11-s + 4.82·13-s + 3·16-s + 1.58·17-s − 1.24·19-s − 3.65·20-s + 0.414·22-s + 7·23-s − 25-s − 1.99·26-s + 5.24·29-s − 5.65·31-s − 4.41·32-s − 0.656·34-s + 7.48·37-s + 0.514·38-s + 3.17·40-s + 6.82·41-s − 11.2·43-s + 1.82·44-s − 2.89·46-s − 4.17·47-s + ⋯ |
L(s) = 1 | − 0.292·2-s − 0.914·4-s + 0.894·5-s + 0.560·8-s − 0.261·10-s − 0.301·11-s + 1.33·13-s + 0.750·16-s + 0.384·17-s − 0.285·19-s − 0.817·20-s + 0.0883·22-s + 1.45·23-s − 0.200·25-s − 0.392·26-s + 0.973·29-s − 1.01·31-s − 0.780·32-s − 0.112·34-s + 1.23·37-s + 0.0834·38-s + 0.501·40-s + 1.06·41-s − 1.71·43-s + 0.275·44-s − 0.427·46-s − 0.608·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.746168209\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.746168209\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 - 7T + 23T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 4.17T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 2.65T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 8.82T + 67T^{2} \) |
| 71 | \( 1 - 9.82T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 5.48T + 97T^{2} \) |
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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
−8.449250601034422348439657264717, −7.72858996474091910668025923090, −6.79521538585397719645147811432, −5.97255803813924492620417901424, −5.41406540298674561710547744502, −4.63414250906547442586579249520, −3.76845253310784830154078976926, −2.90029162951859411235141727478, −1.67635631206165915606167263552, −0.820114828312688323800743213161,
0.820114828312688323800743213161, 1.67635631206165915606167263552, 2.90029162951859411235141727478, 3.76845253310784830154078976926, 4.63414250906547442586579249520, 5.41406540298674561710547744502, 5.97255803813924492620417901424, 6.79521538585397719645147811432, 7.72858996474091910668025923090, 8.449250601034422348439657264717