| L(s) = 1 | + 4-s − 9-s − 11-s + 16-s − 2·31-s − 36-s − 44-s + 49-s − 2·59-s + 64-s + 2·71-s + 81-s + 2·89-s + 99-s + ⋯ |
| L(s) = 1 | + 4-s − 9-s − 11-s + 16-s − 2·31-s − 36-s − 44-s + 49-s − 2·59-s + 64-s + 2·71-s + 81-s + 2·89-s + 99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8416235229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8416235229\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( 1 + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( 1 + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.03665640557783310173742506357, −11.06753192974394098755750756277, −10.56665117242895606014718944826, −9.259365988283614764895460982642, −8.108207639077362766239213111613, −7.29566488255358536017266907596, −6.08870587299178266906232232204, −5.23996740044376597677772147063, −3.38062344130934599018715121049, −2.24118419810646243153124720786,
2.24118419810646243153124720786, 3.38062344130934599018715121049, 5.23996740044376597677772147063, 6.08870587299178266906232232204, 7.29566488255358536017266907596, 8.108207639077362766239213111613, 9.259365988283614764895460982642, 10.56665117242895606014718944826, 11.06753192974394098755750756277, 12.03665640557783310173742506357
