| L(s) = 1 | − 16·17-s − 8·25-s − 16·41-s − 24·49-s − 16·73-s − 64·89-s + 24·97-s − 32·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 3.88·17-s − 8/5·25-s − 2.49·41-s − 3.42·49-s − 1.87·73-s − 6.78·89-s + 2.43·97-s − 3.01·113-s − 1.81·121-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 − 0.307·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 108 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
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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
−6.33634472151797102418282571118, −6.07351728174672932829916405728, −5.92628975608938599128814426589, −5.65991749069864361157359676699, −5.51672857862294356425316622623, −5.19249205268911090698096068860, −5.03245475520656721224281305326, −4.89648651086469713053182223276, −4.71447799647696074306112532320, −4.40932880051237344618820879619, −4.37801765600514754288496063759, −4.02955192529038349022834710434, −3.97160603278328112310309478529, −3.80161291500643957169767468922, −3.39690942768533962921225734363, −3.19404208550402053029823993271, −3.11475485408224554754741143931, −2.66659073004557938944240417495, −2.34457783489642383052106723389, −2.32925154040110143339782242230, −2.31894169013915965413894863647, −1.63655580842767063566560863814, −1.57943344908123840120304227040, −1.37050325363087774214320821124, −1.19096431602155705866920923702, 0, 0, 0, 0,
1.19096431602155705866920923702, 1.37050325363087774214320821124, 1.57943344908123840120304227040, 1.63655580842767063566560863814, 2.31894169013915965413894863647, 2.32925154040110143339782242230, 2.34457783489642383052106723389, 2.66659073004557938944240417495, 3.11475485408224554754741143931, 3.19404208550402053029823993271, 3.39690942768533962921225734363, 3.80161291500643957169767468922, 3.97160603278328112310309478529, 4.02955192529038349022834710434, 4.37801765600514754288496063759, 4.40932880051237344618820879619, 4.71447799647696074306112532320, 4.89648651086469713053182223276, 5.03245475520656721224281305326, 5.19249205268911090698096068860, 5.51672857862294356425316622623, 5.65991749069864361157359676699, 5.92628975608938599128814426589, 6.07351728174672932829916405728, 6.33634472151797102418282571118
