| L(s) = 1 | + 1.83·2-s − 0.364·3-s + 1.36·4-s − 2.83·5-s − 0.668·6-s + 4.39·7-s − 1.16·8-s − 2.86·9-s − 5.19·10-s − 4.39·11-s − 0.497·12-s + 4·13-s + 8.06·14-s + 1.03·15-s − 4.86·16-s + 4.39·17-s − 5.25·18-s + 1.80·19-s − 3.86·20-s − 1.60·21-s − 8.06·22-s + 2.72·23-s + 0.424·24-s + 3.03·25-s + 7.33·26-s + 2.13·27-s + 6.00·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s − 0.210·3-s + 0.682·4-s − 1.26·5-s − 0.272·6-s + 1.66·7-s − 0.412·8-s − 0.955·9-s − 1.64·10-s − 1.32·11-s − 0.143·12-s + 1.10·13-s + 2.15·14-s + 0.266·15-s − 1.21·16-s + 1.06·17-s − 1.23·18-s + 0.413·19-s − 0.864·20-s − 0.349·21-s − 1.71·22-s + 0.569·23-s + 0.0867·24-s + 0.606·25-s + 1.43·26-s + 0.411·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.344997825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.344997825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 - T \) |
| good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 3 | \( 1 + 0.364T + 3T^{2} \) |
| 5 | \( 1 + 2.83T + 5T^{2} \) |
| 7 | \( 1 - 4.39T + 7T^{2} \) |
| 11 | \( 1 + 4.39T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 4.39T + 17T^{2} \) |
| 19 | \( 1 - 1.80T + 19T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 - 2.03T + 29T^{2} \) |
| 31 | \( 1 + 5.66T + 31T^{2} \) |
| 37 | \( 1 - 1.07T + 37T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 3.27T + 47T^{2} \) |
| 53 | \( 1 + 3.66T + 53T^{2} \) |
| 59 | \( 1 + 3.60T + 59T^{2} \) |
| 61 | \( 1 + 8.46T + 61T^{2} \) |
| 67 | \( 1 - 3.33T + 67T^{2} \) |
| 73 | \( 1 - 2.83T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 5.80T + 83T^{2} \) |
| 89 | \( 1 - 0.509T + 89T^{2} \) |
| 97 | \( 1 + 6.06T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.62793798484230881492909966791, −13.74288967463562230500089742443, −12.46184269394296987296090688355, −11.45908044141419367204640202737, −11.01334371457850577799822062059, −8.500711014286350441948373343748, −7.70483926821236134022458458967, −5.64132818777164356787021460477, −4.76166880463423491235826636815, −3.29794053663541918178362444025,
3.29794053663541918178362444025, 4.76166880463423491235826636815, 5.64132818777164356787021460477, 7.70483926821236134022458458967, 8.500711014286350441948373343748, 11.01334371457850577799822062059, 11.45908044141419367204640202737, 12.46184269394296987296090688355, 13.74288967463562230500089742443, 14.62793798484230881492909966791
