| L(s) = 1 | − 1.82·2-s + 2.13·3-s + 1.31·4-s − 4.26·5-s − 3.88·6-s + 7-s + 1.24·8-s + 1.55·9-s + 7.76·10-s + 5.61·11-s + 2.80·12-s + 3.58·13-s − 1.82·14-s − 9.10·15-s − 4.90·16-s + 6.50·17-s − 2.83·18-s + 1.68·19-s − 5.60·20-s + 2.13·21-s − 10.2·22-s + 3.27·23-s + 2.66·24-s + 13.1·25-s − 6.52·26-s − 3.08·27-s + 1.31·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 1.23·3-s + 0.657·4-s − 1.90·5-s − 1.58·6-s + 0.377·7-s + 0.441·8-s + 0.518·9-s + 2.45·10-s + 1.69·11-s + 0.810·12-s + 0.994·13-s − 0.486·14-s − 2.35·15-s − 1.22·16-s + 1.57·17-s − 0.667·18-s + 0.385·19-s − 1.25·20-s + 0.465·21-s − 2.17·22-s + 0.683·23-s + 0.543·24-s + 2.63·25-s − 1.28·26-s − 0.593·27-s + 0.248·28-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
| \(L(1)\) |
\(\approx\) |
\(1.499431273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.499431273\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.82T + 2T^{2} \) |
| 3 | \( 1 - 2.13T + 3T^{2} \) |
| 5 | \( 1 + 4.26T + 5T^{2} \) |
| 11 | \( 1 - 5.61T + 11T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 - 6.50T + 17T^{2} \) |
| 19 | \( 1 - 1.68T + 19T^{2} \) |
| 23 | \( 1 - 3.27T + 23T^{2} \) |
| 29 | \( 1 - 2.92T + 29T^{2} \) |
| 31 | \( 1 + 0.235T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 + 0.257T + 43T^{2} \) |
| 47 | \( 1 - 4.49T + 47T^{2} \) |
| 53 | \( 1 + 1.70T + 53T^{2} \) |
| 59 | \( 1 - 0.207T + 59T^{2} \) |
| 61 | \( 1 - 2.18T + 61T^{2} \) |
| 67 | \( 1 - 0.0456T + 67T^{2} \) |
| 71 | \( 1 - 7.95T + 71T^{2} \) |
| 73 | \( 1 - 3.04T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.298654346545361593096466164812, −7.66858560767977377677781218525, −7.21990859024513644446849728384, −6.38162251182734419845413467346, −4.96776993105839009100066930630, −3.96485247826410109693330487763, −3.70572466566525626489692789842, −2.86122711224704575117066474234, −1.38709429071999978737867854093, −0.869555402140552887783861145537,
0.869555402140552887783861145537, 1.38709429071999978737867854093, 2.86122711224704575117066474234, 3.70572466566525626489692789842, 3.96485247826410109693330487763, 4.96776993105839009100066930630, 6.38162251182734419845413467346, 7.21990859024513644446849728384, 7.66858560767977377677781218525, 8.298654346545361593096466164812
