| L(s) = 1 | + 1.38·2-s − 0.0848·4-s + 4.17·5-s + 3.01·7-s − 2.88·8-s + 5.78·10-s − 11-s + 0.616·13-s + 4.17·14-s − 3.82·16-s + 3.75·17-s + 5.19·19-s − 0.354·20-s − 1.38·22-s + 1.07·23-s + 12.4·25-s + 0.853·26-s − 0.256·28-s + 4.93·29-s − 5.32·31-s + 0.479·32-s + 5.20·34-s + 12.6·35-s − 3.81·37-s + 7.19·38-s − 12.0·40-s − 2.12·41-s + ⋯ |
| L(s) = 1 | + 0.978·2-s − 0.0424·4-s + 1.86·5-s + 1.14·7-s − 1.02·8-s + 1.82·10-s − 0.301·11-s + 0.170·13-s + 1.11·14-s − 0.955·16-s + 0.911·17-s + 1.19·19-s − 0.0792·20-s − 0.295·22-s + 0.223·23-s + 2.49·25-s + 0.167·26-s − 0.0484·28-s + 0.915·29-s − 0.956·31-s + 0.0847·32-s + 0.891·34-s + 2.13·35-s − 0.626·37-s + 1.16·38-s − 1.90·40-s − 0.331·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.156036826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.156036826\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.38T + 2T^{2} \) |
| 5 | \( 1 - 4.17T + 5T^{2} \) |
| 7 | \( 1 - 3.01T + 7T^{2} \) |
| 13 | \( 1 - 0.616T + 13T^{2} \) |
| 17 | \( 1 - 3.75T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 - 4.93T + 29T^{2} \) |
| 31 | \( 1 + 5.32T + 31T^{2} \) |
| 37 | \( 1 + 3.81T + 37T^{2} \) |
| 41 | \( 1 + 2.12T + 41T^{2} \) |
| 43 | \( 1 + 7.97T + 43T^{2} \) |
| 47 | \( 1 + 5.49T + 47T^{2} \) |
| 53 | \( 1 + 1.16T + 53T^{2} \) |
| 59 | \( 1 - 8.82T + 59T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 + 4.50T + 71T^{2} \) |
| 73 | \( 1 - 4.44T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 - 9.81T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
| 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
−8.201232553782764282564515633067, −7.14501402883296089419772129889, −6.38420108690052693282082837002, −5.63051449809168794356357497787, −5.12987055339334374033539808735, −4.90795718578988010831418645254, −3.60936788192127306814045293049, −2.87352946052161014880334005584, −1.94504933889133765558635563598, −1.13741526280303690911020803175,
1.13741526280303690911020803175, 1.94504933889133765558635563598, 2.87352946052161014880334005584, 3.60936788192127306814045293049, 4.90795718578988010831418645254, 5.12987055339334374033539808735, 5.63051449809168794356357497787, 6.38420108690052693282082837002, 7.14501402883296089419772129889, 8.201232553782764282564515633067
