| L(s) = 1 | − 3·4-s − 2·5-s + 9-s + 5·16-s − 2·19-s + 6·20-s − 25-s + 4·29-s + 16·31-s − 3·36-s − 4·41-s − 2·45-s − 14·49-s − 24·59-s − 4·61-s − 3·64-s + 6·76-s − 10·80-s + 81-s − 4·89-s + 4·95-s + 3·100-s − 20·101-s − 20·109-s − 12·116-s − 22·121-s − 48·124-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 0.894·5-s + 1/3·9-s + 5/4·16-s − 0.458·19-s + 1.34·20-s − 1/5·25-s + 0.742·29-s + 2.87·31-s − 1/2·36-s − 0.624·41-s − 0.298·45-s − 2·49-s − 3.12·59-s − 0.512·61-s − 3/8·64-s + 0.688·76-s − 1.11·80-s + 1/9·81-s − 0.423·89-s + 0.410·95-s + 3/10·100-s − 1.99·101-s − 1.91·109-s − 1.11·116-s − 2·121-s − 4.31·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.438835779453266646722697155473, −9.032116632714081125535639739797, −8.263513432862467471567669978548, −8.129458035743720070592981027873, −7.81423371583594124822281317820, −6.83493330235856618687394516740, −6.46787589567348650226556085189, −5.78421312004364849132957140059, −4.89453559736176945278761858504, −4.56991321080597356703354311172, −4.24814179905599513569109101908, −3.44365518362789223021999993571, −2.80168364658006648054788403582, −1.32661394670938609373916514708, 0,
1.32661394670938609373916514708, 2.80168364658006648054788403582, 3.44365518362789223021999993571, 4.24814179905599513569109101908, 4.56991321080597356703354311172, 4.89453559736176945278761858504, 5.78421312004364849132957140059, 6.46787589567348650226556085189, 6.83493330235856618687394516740, 7.81423371583594124822281317820, 8.129458035743720070592981027873, 8.263513432862467471567669978548, 9.032116632714081125535639739797, 9.438835779453266646722697155473
