L(s) = 1 | + 2·4-s − 4·7-s + 2·13-s − 2·19-s − 8·28-s − 2·31-s − 16·37-s − 22·43-s − 2·49-s + 4·52-s + 10·61-s − 8·64-s + 14·67-s − 22·73-s − 4·76-s − 14·79-s − 8·91-s + 14·97-s + 14·103-s − 2·109-s − 16·121-s − 4·124-s + 127-s + 131-s + 8·133-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 4-s − 1.51·7-s + 0.554·13-s − 0.458·19-s − 1.51·28-s − 0.359·31-s − 2.63·37-s − 3.35·43-s − 2/7·49-s + 0.554·52-s + 1.28·61-s − 64-s + 1.71·67-s − 2.57·73-s − 0.458·76-s − 1.57·79-s − 0.838·91-s + 1.42·97-s + 1.37·103-s − 0.191·109-s − 1.45·121-s − 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.693·133-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36905625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36905625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−7.896023416387316924987775456958, −7.24002361353899934476797604702, −7.14530190394130024227769269298, −6.77199008323373875461092611203, −6.60336360365791201562163024207, −6.19893807282492399989648060617, −5.93803341200205461626794735251, −5.46819545625682359071677540787, −4.99508428817044183506351901971, −4.77178676416567706532410830168, −4.10519787553572305042732635609, −3.60675821842974492129722607826, −3.36479820308400257402539502932, −3.14245968042950154191006223597, −2.62051425614622438214464670419, −1.88686163342032147582417204350, −1.87145940004656778494320204254, −1.15587054095223283486260837620, 0, 0,
1.15587054095223283486260837620, 1.87145940004656778494320204254, 1.88686163342032147582417204350, 2.62051425614622438214464670419, 3.14245968042950154191006223597, 3.36479820308400257402539502932, 3.60675821842974492129722607826, 4.10519787553572305042732635609, 4.77178676416567706532410830168, 4.99508428817044183506351901971, 5.46819545625682359071677540787, 5.93803341200205461626794735251, 6.19893807282492399989648060617, 6.60336360365791201562163024207, 6.77199008323373875461092611203, 7.14530190394130024227769269298, 7.24002361353899934476797604702, 7.896023416387316924987775456958