| L(s) = 1 | − 1.68·2-s + 1.73·3-s + 0.826·4-s + 1.52·5-s − 2.91·6-s − 0.782·7-s + 1.97·8-s + 0.0107·9-s − 2.57·10-s + 4.82·11-s + 1.43·12-s + 1.31·13-s + 1.31·14-s + 2.65·15-s − 4.96·16-s + 0.699·17-s − 0.0180·18-s − 3.13·19-s + 1.26·20-s − 1.35·21-s − 8.10·22-s − 3.56·23-s + 3.42·24-s − 2.66·25-s − 2.20·26-s − 5.18·27-s − 0.646·28-s + ⋯ |
| L(s) = 1 | − 1.18·2-s + 1.00·3-s + 0.413·4-s + 0.683·5-s − 1.19·6-s − 0.295·7-s + 0.697·8-s + 0.00358·9-s − 0.812·10-s + 1.45·11-s + 0.414·12-s + 0.363·13-s + 0.351·14-s + 0.684·15-s − 1.24·16-s + 0.169·17-s − 0.00426·18-s − 0.719·19-s + 0.282·20-s − 0.296·21-s − 1.72·22-s − 0.743·23-s + 0.698·24-s − 0.532·25-s − 0.432·26-s − 0.998·27-s − 0.122·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8094377694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8094377694\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.68T + 2T^{2} \) |
| 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 - 1.52T + 5T^{2} \) |
| 7 | \( 1 + 0.782T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 1.31T + 13T^{2} \) |
| 17 | \( 1 - 0.699T + 17T^{2} \) |
| 19 | \( 1 + 3.13T + 19T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 - 0.647T + 29T^{2} \) |
| 31 | \( 1 + 9.09T + 31T^{2} \) |
| 37 | \( 1 + 1.31T + 37T^{2} \) |
| 41 | \( 1 - 1.14T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 5.79T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 4.01T + 59T^{2} \) |
| 61 | \( 1 - 0.673T + 61T^{2} \) |
| 67 | \( 1 - 6.65T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 - 3.39T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 4.30T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−14.00993148272635079692171410673, −12.98251969308015415281354630774, −11.42181615698914700114732652696, −10.13871965579576753293199788422, −9.224881655235906053461356907210, −8.745745886075653360634373781944, −7.52353986627829083931369752808, −6.11702007608492173351411256556, −3.88663656259377248157834601651, −1.90754576601330766553097692582,
1.90754576601330766553097692582, 3.88663656259377248157834601651, 6.11702007608492173351411256556, 7.52353986627829083931369752808, 8.745745886075653360634373781944, 9.224881655235906053461356907210, 10.13871965579576753293199788422, 11.42181615698914700114732652696, 12.98251969308015415281354630774, 14.00993148272635079692171410673
