| L(s) = 1 | − 2.46·2-s + 3.35·3-s + 4.07·4-s − 4.25·5-s − 8.27·6-s + 7-s − 5.12·8-s + 8.24·9-s + 10.5·10-s − 5.52·11-s + 13.6·12-s + 3.47·13-s − 2.46·14-s − 14.2·15-s + 4.48·16-s − 2.75·17-s − 20.3·18-s − 1.06·19-s − 17.3·20-s + 3.35·21-s + 13.6·22-s + 2.80·23-s − 17.2·24-s + 13.1·25-s − 8.57·26-s + 17.6·27-s + 4.07·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 1.93·3-s + 2.03·4-s − 1.90·5-s − 3.37·6-s + 0.377·7-s − 1.81·8-s + 2.74·9-s + 3.32·10-s − 1.66·11-s + 3.95·12-s + 0.964·13-s − 0.658·14-s − 3.68·15-s + 1.12·16-s − 0.668·17-s − 4.79·18-s − 0.244·19-s − 3.88·20-s + 0.731·21-s + 2.90·22-s + 0.584·23-s − 3.51·24-s + 2.62·25-s − 1.68·26-s + 3.38·27-s + 0.771·28-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
| \(L(1)\) |
\(\approx\) |
\(1.117163202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.117163202\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 - 3.35T + 3T^{2} \) |
| 5 | \( 1 + 4.25T + 5T^{2} \) |
| 11 | \( 1 + 5.52T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 2.75T + 17T^{2} \) |
| 19 | \( 1 + 1.06T + 19T^{2} \) |
| 23 | \( 1 - 2.80T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 - 8.67T + 31T^{2} \) |
| 37 | \( 1 + 9.07T + 37T^{2} \) |
| 41 | \( 1 - 0.671T + 41T^{2} \) |
| 43 | \( 1 + 4.24T + 43T^{2} \) |
| 47 | \( 1 + 6.80T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 5.20T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 8.04T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 9.91T + 79T^{2} \) |
| 83 | \( 1 - 3.31T + 83T^{2} \) |
| 89 | \( 1 - 6.18T + 89T^{2} \) |
| 97 | \( 1 + 9.83T + 97T^{2} \) |
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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
−8.194847786029873841632454080442, −7.87671739438926695592127686364, −7.16304726152829648471761708838, −6.75016741400400639079442286239, −4.91710552559585226210408801023, −4.11745393939400597312670984605, −3.25984268787372513867542606369, −2.70826127509306881214256857919, −1.78302591925699397859454668228, −0.65179333746160095425411351226,
0.65179333746160095425411351226, 1.78302591925699397859454668228, 2.70826127509306881214256857919, 3.25984268787372513867542606369, 4.11745393939400597312670984605, 4.91710552559585226210408801023, 6.75016741400400639079442286239, 7.16304726152829648471761708838, 7.87671739438926695592127686364, 8.194847786029873841632454080442
