| L(s) = 1 | − 3·9-s − 18·11-s − 6·19-s − 6·31-s − 24·41-s − 2·49-s + 6·59-s − 42·61-s − 2·79-s + 18·89-s + 54·99-s − 18·101-s − 34·109-s + 167·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 18·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 9-s − 5.42·11-s − 1.37·19-s − 1.07·31-s − 3.74·41-s − 2/7·49-s + 0.781·59-s − 5.37·61-s − 0.225·79-s + 1.90·89-s + 5.42·99-s − 1.79·101-s − 3.25·109-s + 15.1·121-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 + 1.37·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 19 T^{2} - 168 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 85 T^{2} + 5016 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 79 T^{2} + 3432 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 21 T + 208 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + T^{2} - 5328 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.08768857594406002828074417605, −6.38557558593624985907956757284, −6.38485460835652890825387728369, −6.27358226085314040819139179374, −6.20938857716596595247317117082, −5.64359965940695712022344153467, −5.51377896887115323625019517544, −5.38044841727998486059698305274, −5.29924512609774097012246826055, −4.96469154143409362372008525671, −4.84042675216014226789944528880, −4.82852255630888881980074590133, −4.44606746361621602419797657604, −4.17011460602346200596067350531, −3.74869272563242552627648138856, −3.47028954183801285644802214721, −3.34297855185299491499346568796, −2.96883282160106709929240573244, −2.91387404381354825588418963718, −2.61787826604039385373127406938, −2.37983018282427193253100459633, −2.22114959563591971743055697786, −1.99052856617929217683778745171, −1.43744347115331813429196001157, −1.27165119230246777618525923499, 0, 0, 0, 0,
1.27165119230246777618525923499, 1.43744347115331813429196001157, 1.99052856617929217683778745171, 2.22114959563591971743055697786, 2.37983018282427193253100459633, 2.61787826604039385373127406938, 2.91387404381354825588418963718, 2.96883282160106709929240573244, 3.34297855185299491499346568796, 3.47028954183801285644802214721, 3.74869272563242552627648138856, 4.17011460602346200596067350531, 4.44606746361621602419797657604, 4.82852255630888881980074590133, 4.84042675216014226789944528880, 4.96469154143409362372008525671, 5.29924512609774097012246826055, 5.38044841727998486059698305274, 5.51377896887115323625019517544, 5.64359965940695712022344153467, 6.20938857716596595247317117082, 6.27358226085314040819139179374, 6.38485460835652890825387728369, 6.38557558593624985907956757284, 7.08768857594406002828074417605
