| L(s) = 1 | − 4·5-s + 4·9-s + 8·13-s + 10·25-s − 16·45-s + 10·49-s + 40·61-s − 32·65-s − 6·81-s − 24·101-s + 72·113-s + 32·117-s − 44·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 4/3·9-s + 2.21·13-s + 2·25-s − 2.38·45-s + 10/7·49-s + 5.12·61-s − 3.96·65-s − 2/3·81-s − 2.38·101-s + 6.77·113-s + 2.95·117-s − 4·121-s − 1.78·125-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 − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.851139991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.851139991\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.63874126583953474137638714554, −6.14609225243367641491575890041, −6.00865218586950210771019641746, −5.75748705926315709846008956760, −5.72949131297237132898875317395, −5.35435712921148155507260270821, −5.05321482325895381641460440560, −4.85717089262030699813111390975, −4.75656397615886530480565881498, −4.45567589124835971058304903796, −4.08296959178421843427406268782, −3.94264044781964025811498683169, −3.84201259040925531503723434117, −3.77318123122163781428604160620, −3.48567451272554307192641393452, −3.35346738865673206052105529074, −2.72279334321926800202505893963, −2.70343574448862352249166765533, −2.42747041817067414949177692748, −2.01307209535874782893143414303, −1.62660092821968019710734016055, −1.32349232062296915615677684455, −1.08102658257922224359603649971, −0.73735364301492171204219814332, −0.41918521850967959061770407948,
0.41918521850967959061770407948, 0.73735364301492171204219814332, 1.08102658257922224359603649971, 1.32349232062296915615677684455, 1.62660092821968019710734016055, 2.01307209535874782893143414303, 2.42747041817067414949177692748, 2.70343574448862352249166765533, 2.72279334321926800202505893963, 3.35346738865673206052105529074, 3.48567451272554307192641393452, 3.77318123122163781428604160620, 3.84201259040925531503723434117, 3.94264044781964025811498683169, 4.08296959178421843427406268782, 4.45567589124835971058304903796, 4.75656397615886530480565881498, 4.85717089262030699813111390975, 5.05321482325895381641460440560, 5.35435712921148155507260270821, 5.72949131297237132898875317395, 5.75748705926315709846008956760, 6.00865218586950210771019641746, 6.14609225243367641491575890041, 6.63874126583953474137638714554
