| L(s) = 1 | + 50·3-s − 218·5-s + 614·7-s + 1.77e3·9-s − 2.19e3·13-s − 1.09e4·15-s + 3.17e3·17-s + 3.07e4·21-s + 3.18e4·25-s + 5.21e4·27-s + 2.78e4·31-s − 1.33e5·35-s + 1.38e4·37-s − 1.09e5·39-s + 1.11e5·43-s − 3.86e5·45-s − 1.28e5·47-s + 2.59e5·49-s + 1.58e5·51-s + 1.08e6·63-s + 4.78e5·65-s + 3.17e5·71-s + 1.59e6·75-s + 1.31e6·81-s − 6.91e5·85-s − 1.34e6·91-s + 1.39e6·93-s + ⋯ |
| L(s) = 1 | + 1.85·3-s − 1.74·5-s + 1.79·7-s + 2.42·9-s − 13-s − 3.22·15-s + 0.645·17-s + 3.31·21-s + 2.04·25-s + 2.64·27-s + 0.934·31-s − 3.12·35-s + 0.274·37-s − 1.85·39-s + 1.40·43-s − 4.23·45-s − 1.23·47-s + 2.20·49-s + 1.19·51-s + 4.34·63-s + 1.74·65-s + 0.888·71-s + 3.78·75-s + 2.47·81-s − 1.12·85-s − 1.79·91-s + 1.72·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(4.290645238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.290645238\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 - 50 T + p^{6} T^{2} \) |
| 5 | \( 1 + 218 T + p^{6} T^{2} \) |
| 7 | \( 1 - 614 T + p^{6} T^{2} \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( 1 - 3170 T + p^{6} T^{2} \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( 1 - 27830 T + p^{6} T^{2} \) |
| 37 | \( 1 - 13894 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( 1 - 111490 T + p^{6} T^{2} \) |
| 47 | \( 1 + 128554 T + p^{6} T^{2} \) |
| 53 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( 1 - 317990 T + p^{6} T^{2} \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 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
−10.09555616764959933928864337360, −8.929785115982116395420764856246, −8.141887735203654408895953678611, −7.79779067574422377097957391923, −7.19746986405138588844385494713, −4.82768396846223968724242945694, −4.25792522684558799032857330418, −3.25697536663107794959482148636, −2.20136466490286399083650286051, −0.971924218133457298629039723403,
0.971924218133457298629039723403, 2.20136466490286399083650286051, 3.25697536663107794959482148636, 4.25792522684558799032857330418, 4.82768396846223968724242945694, 7.19746986405138588844385494713, 7.79779067574422377097957391923, 8.141887735203654408895953678611, 8.929785115982116395420764856246, 10.09555616764959933928864337360
