| L(s) = 1 | − 2.65·2-s − 2.99·3-s + 5.04·4-s − 2.08·5-s + 7.95·6-s − 1.11·7-s − 8.08·8-s + 5.97·9-s + 5.52·10-s + 3.38·11-s − 15.1·12-s − 3.23·13-s + 2.96·14-s + 6.23·15-s + 11.3·16-s + 4.29·17-s − 15.8·18-s + 0.715·19-s − 10.4·20-s + 3.34·21-s − 8.98·22-s + 0.981·23-s + 24.2·24-s − 0.671·25-s + 8.59·26-s − 8.89·27-s − 5.62·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 1.72·3-s + 2.52·4-s − 0.930·5-s + 3.24·6-s − 0.421·7-s − 2.85·8-s + 1.99·9-s + 1.74·10-s + 1.02·11-s − 4.36·12-s − 0.897·13-s + 0.791·14-s + 1.60·15-s + 2.84·16-s + 1.04·17-s − 3.73·18-s + 0.164·19-s − 2.34·20-s + 0.729·21-s − 1.91·22-s + 0.204·23-s + 4.94·24-s − 0.134·25-s + 1.68·26-s − 1.71·27-s − 1.06·28-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 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 + 2.99T + 3T^{2} \) |
| 5 | \( 1 + 2.08T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 4.29T + 17T^{2} \) |
| 19 | \( 1 - 0.715T + 19T^{2} \) |
| 23 | \( 1 - 0.981T + 23T^{2} \) |
| 29 | \( 1 + 2.22T + 29T^{2} \) |
| 31 | \( 1 - 0.355T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 1.13T + 41T^{2} \) |
| 43 | \( 1 + 6.13T + 43T^{2} \) |
| 47 | \( 1 + 2.54T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 + 9.28T + 67T^{2} \) |
| 71 | \( 1 + 3.24T + 71T^{2} \) |
| 73 | \( 1 + 4.55T + 73T^{2} \) |
| 79 | \( 1 + 2.45T + 79T^{2} \) |
| 83 | \( 1 - 7.49T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
| 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
−7.68129206606107186409089084960, −7.12105577743389990076655208163, −6.52506671036979906413293193550, −5.97694276984673170817422158572, −5.05881280248955248549508885154, −4.04722785558020424213668100231, −3.03488618240444469014331907768, −1.64592786974979312212520740141, −0.803470463471443516605543451320, 0,
0.803470463471443516605543451320, 1.64592786974979312212520740141, 3.03488618240444469014331907768, 4.04722785558020424213668100231, 5.05881280248955248549508885154, 5.97694276984673170817422158572, 6.52506671036979906413293193550, 7.12105577743389990076655208163, 7.68129206606107186409089084960
