| L(s) = 1 | − 2.79·2-s − 1.21·3-s + 5.80·4-s − 0.914·5-s + 3.38·6-s − 4.48·7-s − 10.6·8-s − 1.53·9-s + 2.55·10-s − 0.426·11-s − 7.04·12-s + 1.28·13-s + 12.5·14-s + 1.10·15-s + 18.1·16-s + 0.227·17-s + 4.27·18-s − 1.98·19-s − 5.31·20-s + 5.43·21-s + 1.19·22-s + 4.20·23-s + 12.9·24-s − 4.16·25-s − 3.59·26-s + 5.49·27-s − 26.0·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s − 0.699·3-s + 2.90·4-s − 0.409·5-s + 1.38·6-s − 1.69·7-s − 3.76·8-s − 0.510·9-s + 0.808·10-s − 0.128·11-s − 2.03·12-s + 0.356·13-s + 3.34·14-s + 0.286·15-s + 4.53·16-s + 0.0552·17-s + 1.00·18-s − 0.454·19-s − 1.18·20-s + 1.18·21-s + 0.254·22-s + 0.876·23-s + 2.63·24-s − 0.832·25-s − 0.704·26-s + 1.05·27-s − 4.91·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 6047 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 3 | \( 1 + 1.21T + 3T^{2} \) |
| 5 | \( 1 + 0.914T + 5T^{2} \) |
| 7 | \( 1 + 4.48T + 7T^{2} \) |
| 11 | \( 1 + 0.426T + 11T^{2} \) |
| 13 | \( 1 - 1.28T + 13T^{2} \) |
| 17 | \( 1 - 0.227T + 17T^{2} \) |
| 19 | \( 1 + 1.98T + 19T^{2} \) |
| 23 | \( 1 - 4.20T + 23T^{2} \) |
| 29 | \( 1 - 6.36T + 29T^{2} \) |
| 31 | \( 1 + 5.34T + 31T^{2} \) |
| 37 | \( 1 + 3.57T + 37T^{2} \) |
| 41 | \( 1 + 3.75T + 41T^{2} \) |
| 43 | \( 1 + 7.55T + 43T^{2} \) |
| 47 | \( 1 + 8.56T + 47T^{2} \) |
| 53 | \( 1 + 5.56T + 53T^{2} \) |
| 59 | \( 1 - 0.103T + 59T^{2} \) |
| 61 | \( 1 - 2.40T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 1.33T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 8.22T + 83T^{2} \) |
| 89 | \( 1 - 9.10T + 89T^{2} \) |
| 97 | \( 1 + 3.49T + 97T^{2} \) |
| 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
−7.953216848311374304990611805673, −6.82543670127546970474615939227, −6.70397782666872726408096064696, −6.03856432036909375834367828996, −5.21958485102125505745340753109, −3.49907752130144866887128217083, −3.09070161695843887547586295753, −2.03201288640020839918949595729, −0.71580024949487865509808949944, 0,
0.71580024949487865509808949944, 2.03201288640020839918949595729, 3.09070161695843887547586295753, 3.49907752130144866887128217083, 5.21958485102125505745340753109, 6.03856432036909375834367828996, 6.70397782666872726408096064696, 6.82543670127546970474615939227, 7.953216848311374304990611805673
