| L(s) = 1 | − 2-s − 0.347·3-s + 4-s + 0.347·6-s − 1.87·7-s − 8-s − 0.879·9-s − 0.347·12-s − 1.53·13-s + 1.87·14-s + 16-s + 1.53·17-s + 0.879·18-s − 19-s + 0.652·21-s + 0.347·23-s + 0.347·24-s + 1.53·26-s + 0.652·27-s − 1.87·28-s − 1.53·29-s − 32-s − 1.53·34-s − 0.879·36-s + 37-s + 38-s + 0.532·39-s + ⋯ |
| L(s) = 1 | − 2-s − 0.347·3-s + 4-s + 0.347·6-s − 1.87·7-s − 8-s − 0.879·9-s − 0.347·12-s − 1.53·13-s + 1.87·14-s + 16-s + 1.53·17-s + 0.879·18-s − 19-s + 0.652·21-s + 0.347·23-s + 0.347·24-s + 1.53·26-s + 0.652·27-s − 1.87·28-s − 1.53·29-s − 32-s − 1.53·34-s − 0.879·36-s + 37-s + 38-s + 0.532·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3297888109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3297888109\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.347T + T^{2} \) |
| 7 | \( 1 + 1.87T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.53T + T^{2} \) |
| 17 | \( 1 - 1.53T + T^{2} \) |
| 23 | \( 1 - 0.347T + T^{2} \) |
| 29 | \( 1 + 1.53T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 - 1.87T + T^{2} \) |
| 59 | \( 1 + 0.347T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.347T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
| 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
−8.831655454595356350558178077336, −7.920433166717209397823752313459, −7.22259940738282807331510472649, −6.57910072727121896937510160842, −5.87858761736417789078448667577, −5.29325736723530133678251333539, −3.74698871582306586024589522387, −2.97525884109578343826211530157, −2.29358438037025803274176720683, −0.53475195151724166465600046100,
0.53475195151724166465600046100, 2.29358438037025803274176720683, 2.97525884109578343826211530157, 3.74698871582306586024589522387, 5.29325736723530133678251333539, 5.87858761736417789078448667577, 6.57910072727121896937510160842, 7.22259940738282807331510472649, 7.920433166717209397823752313459, 8.831655454595356350558178077336
