| L(s) = 1 | − 8·9-s + 10·25-s + 48·41-s − 8·49-s + 30·81-s + 24·89-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 80·225-s + ⋯ |
| L(s) = 1 | − 8/3·9-s + 2·25-s + 7.49·41-s − 8/7·49-s + 10/3·81-s + 2.54·89-s − 4·121-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 + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s − 5.33·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.915561420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.915561420\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.96529908559581923470473206272, −6.35867473417400326667114554592, −6.33202653307794440330219547185, −6.31788596532155351043484214172, −6.15379024255503077825596239705, −5.84173369858938056260622045457, −5.55366799139059837885331082955, −5.35570244715184207686346759183, −5.24452195390248714795373551901, −4.97326978725078764107098888676, −4.68638538226063907994768758053, −4.38500888007087595172946702715, −4.11638414089062825535332761058, −3.92763752514989788616977354505, −3.81148480680400456797114359294, −3.05040635433004808333399818227, −3.01490255900182227104995763061, −3.00573131074610767724235944884, −2.60175771206566757867170950348, −2.41223712464075483569710741239, −2.18544267703346915664342748151, −1.62362903636296284698723335247, −1.08467039362434501894254952975, −0.68658930390005499480121379532, −0.52184243415511880416879260795,
0.52184243415511880416879260795, 0.68658930390005499480121379532, 1.08467039362434501894254952975, 1.62362903636296284698723335247, 2.18544267703346915664342748151, 2.41223712464075483569710741239, 2.60175771206566757867170950348, 3.00573131074610767724235944884, 3.01490255900182227104995763061, 3.05040635433004808333399818227, 3.81148480680400456797114359294, 3.92763752514989788616977354505, 4.11638414089062825535332761058, 4.38500888007087595172946702715, 4.68638538226063907994768758053, 4.97326978725078764107098888676, 5.24452195390248714795373551901, 5.35570244715184207686346759183, 5.55366799139059837885331082955, 5.84173369858938056260622045457, 6.15379024255503077825596239705, 6.31788596532155351043484214172, 6.33202653307794440330219547185, 6.35867473417400326667114554592, 6.96529908559581923470473206272
