| L(s) = 1 | − 2.81·2-s + 0.225·3-s + 5.91·4-s − 1.24·5-s − 0.634·6-s + 7-s − 11.0·8-s − 2.94·9-s + 3.50·10-s + 4.12·11-s + 1.33·12-s + 3.68·13-s − 2.81·14-s − 0.280·15-s + 19.1·16-s − 6.21·17-s + 8.29·18-s − 1.83·19-s − 7.36·20-s + 0.225·21-s − 11.6·22-s − 7.83·23-s − 2.48·24-s − 3.44·25-s − 10.3·26-s − 1.34·27-s + 5.91·28-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 0.130·3-s + 2.95·4-s − 0.556·5-s − 0.259·6-s + 0.377·7-s − 3.89·8-s − 0.983·9-s + 1.10·10-s + 1.24·11-s + 0.385·12-s + 1.02·13-s − 0.752·14-s − 0.0725·15-s + 4.79·16-s − 1.50·17-s + 1.95·18-s − 0.420·19-s − 1.64·20-s + 0.0492·21-s − 2.47·22-s − 1.63·23-s − 0.507·24-s − 0.689·25-s − 2.03·26-s − 0.258·27-s + 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4628281420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4628281420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
| good | 2 | \( 1 + 2.81T + 2T^{2} \) |
| 3 | \( 1 - 0.225T + 3T^{2} \) |
| 5 | \( 1 + 1.24T + 5T^{2} \) |
| 11 | \( 1 - 4.12T + 11T^{2} \) |
| 13 | \( 1 - 3.68T + 13T^{2} \) |
| 17 | \( 1 + 6.21T + 17T^{2} \) |
| 19 | \( 1 + 1.83T + 19T^{2} \) |
| 23 | \( 1 + 7.83T + 23T^{2} \) |
| 29 | \( 1 + 9.64T + 29T^{2} \) |
| 31 | \( 1 - 6.68T + 31T^{2} \) |
| 37 | \( 1 - 4.67T + 37T^{2} \) |
| 41 | \( 1 + 5.34T + 41T^{2} \) |
| 43 | \( 1 + 6.30T + 43T^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 - 8.21T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 3.40T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 0.0538T + 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 9.62T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.359611372762010168729452151963, −7.71541183078324988053742221867, −6.80754394636618372666292857118, −6.28708356938679226902272812168, −5.72801592021403773898454679582, −4.09830909709136641002120086211, −3.46035274350085959021701918386, −2.25551437031371794632724471760, −1.71574881776738555166823220220, −0.46763968705885939791629985327,
0.46763968705885939791629985327, 1.71574881776738555166823220220, 2.25551437031371794632724471760, 3.46035274350085959021701918386, 4.09830909709136641002120086211, 5.72801592021403773898454679582, 6.28708356938679226902272812168, 6.80754394636618372666292857118, 7.71541183078324988053742221867, 8.359611372762010168729452151963
