| L(s) = 1 | − 2.06·2-s − 2.80·3-s + 2.26·4-s − 1.02·5-s + 5.79·6-s + 7-s − 0.545·8-s + 4.87·9-s + 2.11·10-s − 1.69·11-s − 6.35·12-s + 0.918·13-s − 2.06·14-s + 2.87·15-s − 3.40·16-s − 3.54·17-s − 10.0·18-s + 0.506·19-s − 2.31·20-s − 2.80·21-s + 3.49·22-s + 2.15·23-s + 1.53·24-s − 3.95·25-s − 1.89·26-s − 5.26·27-s + 2.26·28-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 1.62·3-s + 1.13·4-s − 0.457·5-s + 2.36·6-s + 0.377·7-s − 0.192·8-s + 1.62·9-s + 0.668·10-s − 0.510·11-s − 1.83·12-s + 0.254·13-s − 0.551·14-s + 0.741·15-s − 0.850·16-s − 0.860·17-s − 2.37·18-s + 0.116·19-s − 0.518·20-s − 0.612·21-s + 0.744·22-s + 0.448·23-s + 0.312·24-s − 0.790·25-s − 0.371·26-s − 1.01·27-s + 0.427·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2095113258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2095113258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 859 | \( 1 + T \) |
| good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 1.02T + 5T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 - 0.918T + 13T^{2} \) |
| 17 | \( 1 + 3.54T + 17T^{2} \) |
| 19 | \( 1 - 0.506T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 + 0.298T + 29T^{2} \) |
| 31 | \( 1 + 3.50T + 31T^{2} \) |
| 37 | \( 1 + 6.21T + 37T^{2} \) |
| 41 | \( 1 - 0.683T + 41T^{2} \) |
| 43 | \( 1 - 2.55T + 43T^{2} \) |
| 47 | \( 1 + 8.23T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 8.98T + 59T^{2} \) |
| 61 | \( 1 + 0.678T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 9.74T + 71T^{2} \) |
| 73 | \( 1 - 7.51T + 73T^{2} \) |
| 79 | \( 1 + 9.35T + 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 + 6.72T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.117211976294998139049681758751, −7.31468294712697497114508435321, −6.92045149417967237940721775871, −6.09381290119660826974705614086, −5.29516406182467270198304495799, −4.66597364649763768418956988708, −3.79447535068853562546131555029, −2.28985838917579193624479586829, −1.34010631051603406888583283820, −0.35827317074151567140911710597,
0.35827317074151567140911710597, 1.34010631051603406888583283820, 2.28985838917579193624479586829, 3.79447535068853562546131555029, 4.66597364649763768418956988708, 5.29516406182467270198304495799, 6.09381290119660826974705614086, 6.92045149417967237940721775871, 7.31468294712697497114508435321, 8.117211976294998139049681758751
