| L(s) = 1 | − 2·4-s − 4·7-s + 4·16-s + 8·28-s − 20·31-s − 2·49-s − 8·64-s − 28·73-s − 20·79-s − 4·97-s − 28·103-s − 16·112-s − 10·121-s + 40·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 4-s − 1.51·7-s + 16-s + 1.51·28-s − 3.59·31-s − 2/7·49-s − 64-s − 3.27·73-s − 2.25·79-s − 0.406·97-s − 2.75·103-s − 1.51·112-s − 0.909·121-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.097278333646016043644896632751, −8.947174569086627530166685160944, −8.186389872541100225267746582602, −8.159503348502677388655341050829, −7.27555897520815826322214297244, −7.16980034059035337689784753713, −6.87201030730388695291060964067, −5.97841109450943491439673499190, −5.95937510888850869399184321810, −5.48695190598253148257586371512, −5.08098196072103224937510906616, −4.27738163237587234390225253332, −4.22532595064622352166774962568, −3.38481122187681106371870946495, −3.36275080870364268618296400089, −2.76014854368415329163206817809, −1.87994880808488266813241989632, −1.30825775781324601032672761017, 0, 0,
1.30825775781324601032672761017, 1.87994880808488266813241989632, 2.76014854368415329163206817809, 3.36275080870364268618296400089, 3.38481122187681106371870946495, 4.22532595064622352166774962568, 4.27738163237587234390225253332, 5.08098196072103224937510906616, 5.48695190598253148257586371512, 5.95937510888850869399184321810, 5.97841109450943491439673499190, 6.87201030730388695291060964067, 7.16980034059035337689784753713, 7.27555897520815826322214297244, 8.159503348502677388655341050829, 8.186389872541100225267746582602, 8.947174569086627530166685160944, 9.097278333646016043644896632751
