| L(s) = 1 | − 2.79·2-s − 0.951·3-s + 5.83·4-s + 2.72·5-s + 2.66·6-s + 1.55·7-s − 10.7·8-s − 2.09·9-s − 7.62·10-s + 4.52·11-s − 5.55·12-s − 4.81·13-s − 4.36·14-s − 2.59·15-s + 18.3·16-s + 4.93·17-s + 5.86·18-s − 6.45·19-s + 15.8·20-s − 1.48·21-s − 12.6·22-s − 0.588·23-s + 10.2·24-s + 2.42·25-s + 13.4·26-s + 4.84·27-s + 9.09·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s − 0.549·3-s + 2.91·4-s + 1.21·5-s + 1.08·6-s + 0.589·7-s − 3.79·8-s − 0.698·9-s − 2.41·10-s + 1.36·11-s − 1.60·12-s − 1.33·13-s − 1.16·14-s − 0.669·15-s + 4.59·16-s + 1.19·17-s + 1.38·18-s − 1.48·19-s + 3.55·20-s − 0.323·21-s − 2.69·22-s − 0.122·23-s + 2.08·24-s + 0.484·25-s + 2.64·26-s + 0.933·27-s + 1.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 3 | \( 1 + 0.951T + 3T^{2} \) |
| 5 | \( 1 - 2.72T + 5T^{2} \) |
| 7 | \( 1 - 1.55T + 7T^{2} \) |
| 11 | \( 1 - 4.52T + 11T^{2} \) |
| 13 | \( 1 + 4.81T + 13T^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 23 | \( 1 + 0.588T + 23T^{2} \) |
| 29 | \( 1 + 7.40T + 29T^{2} \) |
| 31 | \( 1 - 4.71T + 31T^{2} \) |
| 37 | \( 1 + 9.59T + 37T^{2} \) |
| 41 | \( 1 - 3.80T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 + 2.11T + 47T^{2} \) |
| 53 | \( 1 - 7.56T + 53T^{2} \) |
| 59 | \( 1 + 0.485T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 1.53T + 67T^{2} \) |
| 73 | \( 1 - 9.26T + 73T^{2} \) |
| 79 | \( 1 - 1.69T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 4.19T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
| 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
−7.58365649519988164450631784118, −6.86773805947562672828478729338, −6.32379124891580767370550855313, −5.74670631462793284851761215215, −5.08649452327582598418175527428, −3.59321194048084755166330415633, −2.46031723236294319896749183835, −1.93701637100923623817843667561, −1.15869492755580855083213079240, 0,
1.15869492755580855083213079240, 1.93701637100923623817843667561, 2.46031723236294319896749183835, 3.59321194048084755166330415633, 5.08649452327582598418175527428, 5.74670631462793284851761215215, 6.32379124891580767370550855313, 6.86773805947562672828478729338, 7.58365649519988164450631784118
