| L(s) = 1 | − 4·5-s + 10·25-s + 24·37-s + 16·41-s + 16·43-s + 8·47-s + 6·49-s − 24·59-s + 40·83-s − 8·89-s − 16·101-s + 40·121-s − 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s − 96·185-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 2·25-s + 3.94·37-s + 2.49·41-s + 2.43·43-s + 1.16·47-s + 6/7·49-s − 3.12·59-s + 4.39·83-s − 0.847·89-s − 1.59·101-s + 3.63·121-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 + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s − 7.05·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.877128085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.877128085\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 454 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1318 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - 1198 T^{4} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4342 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 8434 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 11638 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 200 T^{2} + 19762 T^{4} - 200 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 7306 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 20 T + 246 T^{2} - 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 16426 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.08471910616495079392157428603, −6.68820084606764138739898179121, −6.30752486149898045194387278665, −6.25547621312898429127110977670, −6.23186112287728978611856607465, −5.79220033243376526133002388523, −5.63636199517904002775308532236, −5.49101894215325531795926211712, −5.04882602754064769915934492739, −4.71772800988564053240207090529, −4.50611456636985345297679076317, −4.38138444898886458203700803940, −4.26918156955569299867081174388, −3.98661509134484489436883332407, −3.79930490060728495644405467371, −3.35124525566485841973865349282, −3.28903739640957799550243330798, −2.63033718786387273821823845328, −2.61693556462327820932714466230, −2.61128972299291195155790488774, −2.07457981900935902566199092608, −1.54152698799098190199158213886, −0.991728723398788415882534976279, −0.74017769634497723793396981006, −0.55204498418471840298526922929,
0.55204498418471840298526922929, 0.74017769634497723793396981006, 0.991728723398788415882534976279, 1.54152698799098190199158213886, 2.07457981900935902566199092608, 2.61128972299291195155790488774, 2.61693556462327820932714466230, 2.63033718786387273821823845328, 3.28903739640957799550243330798, 3.35124525566485841973865349282, 3.79930490060728495644405467371, 3.98661509134484489436883332407, 4.26918156955569299867081174388, 4.38138444898886458203700803940, 4.50611456636985345297679076317, 4.71772800988564053240207090529, 5.04882602754064769915934492739, 5.49101894215325531795926211712, 5.63636199517904002775308532236, 5.79220033243376526133002388523, 6.23186112287728978611856607465, 6.25547621312898429127110977670, 6.30752486149898045194387278665, 6.68820084606764138739898179121, 7.08471910616495079392157428603
