| L(s) = 1 | + 4·13-s + 14·25-s − 16·37-s + 10·49-s − 28·61-s + 16·73-s + 4·97-s + 16·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.10·13-s + 14/5·25-s − 2.63·37-s + 10/7·49-s − 3.58·61-s + 1.87·73-s + 0.406·97-s + 1.53·109-s + 0.909·121-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 − 3.23·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.838877859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.838877859\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 77 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 67 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 67 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 139 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 166 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
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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
−6.94610025566856022645148648061, −6.72017047940997203420020949378, −6.48134687865389440413011881851, −6.23462554012693887878561223955, −5.93239024354581407464664249190, −5.89956498832381126244273500963, −5.70383470913063902782134990027, −5.16948438981566945141074046682, −5.14709397107151442089396502369, −4.76587075501338825507301666772, −4.74545984483450127452584457741, −4.62604249616783951873412764168, −4.08912966173216204126720949169, −3.66808566745617160278130961505, −3.66541692634906408004630710595, −3.62078361246027658319344766399, −2.98399805629499848359794861927, −2.89775493735339319923820077677, −2.74081711422851606293951784361, −2.29036642415981658647050751702, −1.81001087703213092257857322970, −1.56518974006172941652756626440, −1.39537144777438267960550015576, −0.75292952346528297811631648082, −0.50023974819583373243254670989,
0.50023974819583373243254670989, 0.75292952346528297811631648082, 1.39537144777438267960550015576, 1.56518974006172941652756626440, 1.81001087703213092257857322970, 2.29036642415981658647050751702, 2.74081711422851606293951784361, 2.89775493735339319923820077677, 2.98399805629499848359794861927, 3.62078361246027658319344766399, 3.66541692634906408004630710595, 3.66808566745617160278130961505, 4.08912966173216204126720949169, 4.62604249616783951873412764168, 4.74545984483450127452584457741, 4.76587075501338825507301666772, 5.14709397107151442089396502369, 5.16948438981566945141074046682, 5.70383470913063902782134990027, 5.89956498832381126244273500963, 5.93239024354581407464664249190, 6.23462554012693887878561223955, 6.48134687865389440413011881851, 6.72017047940997203420020949378, 6.94610025566856022645148648061
