| L(s) = 1 | − 2.77·2-s − 1.97·3-s + 5.72·4-s − 3.34·5-s + 5.50·6-s + 7-s − 10.3·8-s + 0.917·9-s + 9.29·10-s − 6.19·11-s − 11.3·12-s − 4.77·13-s − 2.77·14-s + 6.61·15-s + 17.3·16-s + 6.95·17-s − 2.55·18-s + 5.45·19-s − 19.1·20-s − 1.97·21-s + 17.2·22-s − 5.40·23-s + 20.5·24-s + 6.17·25-s + 13.2·26-s + 4.12·27-s + 5.72·28-s + ⋯ |
| L(s) = 1 | − 1.96·2-s − 1.14·3-s + 2.86·4-s − 1.49·5-s + 2.24·6-s + 0.377·7-s − 3.66·8-s + 0.305·9-s + 2.93·10-s − 1.86·11-s − 3.27·12-s − 1.32·13-s − 0.742·14-s + 1.70·15-s + 4.33·16-s + 1.68·17-s − 0.601·18-s + 1.25·19-s − 4.28·20-s − 0.431·21-s + 3.66·22-s − 1.12·23-s + 4.18·24-s + 1.23·25-s + 2.60·26-s + 0.793·27-s + 1.08·28-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
| \(L(1)\) |
\(\approx\) |
\(0.09049430945\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09049430945\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 + 1.97T + 3T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 11 | \( 1 + 6.19T + 11T^{2} \) |
| 13 | \( 1 + 4.77T + 13T^{2} \) |
| 17 | \( 1 - 6.95T + 17T^{2} \) |
| 19 | \( 1 - 5.45T + 19T^{2} \) |
| 23 | \( 1 + 5.40T + 23T^{2} \) |
| 29 | \( 1 - 1.50T + 29T^{2} \) |
| 31 | \( 1 - 6.57T + 31T^{2} \) |
| 37 | \( 1 - 2.34T + 37T^{2} \) |
| 41 | \( 1 - 3.44T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 - 4.97T + 47T^{2} \) |
| 53 | \( 1 + 0.487T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 7.29T + 67T^{2} \) |
| 71 | \( 1 + 0.563T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 2.20T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 1.00T + 89T^{2} \) |
| 97 | \( 1 + 3.71T + 97T^{2} \) |
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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
−7.998652852571154141424825955825, −7.49437246643061938806334295021, −7.27719305906607171907621561518, −6.06006785226279553356836029974, −5.44851756998014903842043888337, −4.70167643949200785590712560954, −3.16036468425357958855859017373, −2.66135020937128645011309752270, −1.20369486300204629341682208876, −0.25412994986459597967863117513,
0.25412994986459597967863117513, 1.20369486300204629341682208876, 2.66135020937128645011309752270, 3.16036468425357958855859017373, 4.70167643949200785590712560954, 5.44851756998014903842043888337, 6.06006785226279553356836029974, 7.27719305906607171907621561518, 7.49437246643061938806334295021, 7.998652852571154141424825955825
