| L(s) = 1 | + 2·5-s − 2·7-s − 2·9-s − 11-s − 2·19-s + 25-s − 4·35-s − 2·43-s − 4·45-s + 49-s − 2·55-s + 4·63-s + 2·77-s + 3·81-s + 4·83-s − 4·95-s + 2·99-s − 2·125-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2·5-s − 2·7-s − 2·9-s − 11-s − 2·19-s + 25-s − 4·35-s − 2·43-s − 4·45-s + 49-s − 2·55-s + 4·63-s + 2·77-s + 3·81-s + 4·83-s − 4·95-s + 2·99-s − 2·125-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4955163753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4955163753\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$ | \( ( 1 - T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.116124634607260209711976603861, −8.673920470438124244113198324602, −8.391024482195788646960181817374, −7.936182055181733746265555181945, −7.62851729711186109030202678200, −6.78971483632858839388669619393, −6.42290523941451150081624021416, −6.33940024170226370871815005867, −6.22699194579585236291741971785, −5.52868302880481185877540965669, −5.45524093200267442324615891568, −5.03377124109541591979337953843, −4.45711715415669880473255214204, −3.60501805484142871921563506478, −3.45401214877957642521004217867, −2.87163757491949561897780663283, −2.52618364648759361627890946559, −2.16602627131889975193118298740, −1.75515824237564680179202437865, −0.37325347208455571229472566048,
0.37325347208455571229472566048, 1.75515824237564680179202437865, 2.16602627131889975193118298740, 2.52618364648759361627890946559, 2.87163757491949561897780663283, 3.45401214877957642521004217867, 3.60501805484142871921563506478, 4.45711715415669880473255214204, 5.03377124109541591979337953843, 5.45524093200267442324615891568, 5.52868302880481185877540965669, 6.22699194579585236291741971785, 6.33940024170226370871815005867, 6.42290523941451150081624021416, 6.78971483632858839388669619393, 7.62851729711186109030202678200, 7.936182055181733746265555181945, 8.391024482195788646960181817374, 8.673920470438124244113198324602, 9.116124634607260209711976603861
