| L(s) = 1 | − 32·19-s + 52·25-s + 160·43-s + 76·49-s − 320·67-s − 40·73-s + 200·97-s − 164·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 196·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 1.68·19-s + 2.07·25-s + 3.72·43-s + 1.55·49-s − 4.77·67-s − 0.547·73-s + 2.06·97-s − 1.35·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 1.15·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.400606638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.400606638\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 478 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1082 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1862 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 578 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2722 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 2882 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5402 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 6322 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 7202 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 9542 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 5582 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 13282 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{4} \) |
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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
−8.444821048695255887044131700605, −8.116274294181486190511442436369, −7.908385297380946473028717944941, −7.41788989597509426312750939136, −7.32477418899268972819101484298, −7.09214519740534049322799370032, −7.02310083699353109952619861096, −6.32062111031044647527263642304, −6.29560691774140837787138103901, −5.95174786767460069355394983200, −5.85752672140085607083999079693, −5.48219054696703720462176692083, −5.05260582005309430837449462392, −4.75295136240323007128590682680, −4.34211128234409429557935947647, −4.22668848982062245927670690737, −4.19264559819605226257838137509, −3.35620786455081326585926285389, −3.23293859315296355801217254028, −2.78815526408337294670385706546, −2.33380271953532543606523152312, −2.22890814463314777838073640284, −1.45121295747060003886977510261, −1.02359550910949908901703368983, −0.42002331044309771597032873225,
0.42002331044309771597032873225, 1.02359550910949908901703368983, 1.45121295747060003886977510261, 2.22890814463314777838073640284, 2.33380271953532543606523152312, 2.78815526408337294670385706546, 3.23293859315296355801217254028, 3.35620786455081326585926285389, 4.19264559819605226257838137509, 4.22668848982062245927670690737, 4.34211128234409429557935947647, 4.75295136240323007128590682680, 5.05260582005309430837449462392, 5.48219054696703720462176692083, 5.85752672140085607083999079693, 5.95174786767460069355394983200, 6.29560691774140837787138103901, 6.32062111031044647527263642304, 7.02310083699353109952619861096, 7.09214519740534049322799370032, 7.32477418899268972819101484298, 7.41788989597509426312750939136, 7.908385297380946473028717944941, 8.116274294181486190511442436369, 8.444821048695255887044131700605
