L(s) = 1 | − 2.47·2-s + 4.11·4-s − 2.58·5-s − 5.22·8-s + 6.39·10-s + 11-s + 5.87·13-s + 4.70·16-s + 7.51·17-s + 2.35·19-s − 10.6·20-s − 2.47·22-s − 6.94·23-s + 1.69·25-s − 14.5·26-s + 5.87·29-s + 3.66·31-s − 1.16·32-s − 18.5·34-s + 3.30·37-s − 5.83·38-s + 13.5·40-s + 5.28·41-s + 7.40·43-s + 4.11·44-s + 17.1·46-s + 7.53·47-s + ⋯ |
L(s) = 1 | − 1.74·2-s + 2.05·4-s − 1.15·5-s − 1.84·8-s + 2.02·10-s + 0.301·11-s + 1.62·13-s + 1.17·16-s + 1.82·17-s + 0.540·19-s − 2.38·20-s − 0.527·22-s − 1.44·23-s + 0.339·25-s − 2.84·26-s + 1.09·29-s + 0.657·31-s − 0.206·32-s − 3.18·34-s + 0.543·37-s − 0.945·38-s + 2.13·40-s + 0.825·41-s + 1.12·43-s + 0.620·44-s + 2.53·46-s + 1.09·47-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
\(L(1)\) |
\(\approx\) |
\(0.8069395759\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8069395759\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.47T + 2T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 13 | \( 1 - 5.87T + 13T^{2} \) |
| 17 | \( 1 - 7.51T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 + 6.94T + 23T^{2} \) |
| 29 | \( 1 - 5.87T + 29T^{2} \) |
| 31 | \( 1 - 3.66T + 31T^{2} \) |
| 37 | \( 1 - 3.30T + 37T^{2} \) |
| 41 | \( 1 - 5.28T + 41T^{2} \) |
| 43 | \( 1 - 7.40T + 43T^{2} \) |
| 47 | \( 1 - 7.53T + 47T^{2} \) |
| 53 | \( 1 - 4.22T + 53T^{2} \) |
| 59 | \( 1 + 0.926T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 4.45T + 71T^{2} \) |
| 73 | \( 1 - 2.12T + 73T^{2} \) |
| 79 | \( 1 + 4.45T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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
−8.321098273911061139492951626275, −7.68951805629520013587166437526, −7.36023158243649014411385252092, −6.23498923005341219644552925377, −5.81027297059699040529290890965, −4.29103394321137974504671575197, −3.60726508121161199654574817796, −2.69896147832521497530703159070, −1.32841937005603245744345055426, −0.74667800640266865769255345393,
0.74667800640266865769255345393, 1.32841937005603245744345055426, 2.69896147832521497530703159070, 3.60726508121161199654574817796, 4.29103394321137974504671575197, 5.81027297059699040529290890965, 6.23498923005341219644552925377, 7.36023158243649014411385252092, 7.68951805629520013587166437526, 8.321098273911061139492951626275