| L(s) = 1 | − 24·17-s − 4·25-s − 22·49-s − 28·73-s + 24·89-s + 28·97-s − 24·113-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 5.82·17-s − 4/5·25-s − 3.14·49-s − 3.27·73-s + 2.54·89-s + 2.84·97-s − 2.25·113-s − 2.54·121-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 − 0.153·169-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 + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5106767282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5106767282\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 71 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 131 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
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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
−6.57263687288255213501342879076, −6.48740459502075039145385782136, −6.33732903377607424362326405054, −6.13634146391230126780042335600, −5.99096607040410814367549485677, −5.56503337399670877144535030039, −5.17335353397650782263019465923, −5.03126809022214819815043317543, −4.76150642500062643971533485193, −4.74490439064127255540267277504, −4.62871576144711991545922893177, −4.02122885964939440890393802578, −3.97351412925969355109850596586, −3.95127181805860418111835907324, −3.74344345731746729832954448024, −3.00835853466204950898612402687, −2.84561326542897479118525896008, −2.82605179923579814619731568052, −2.37757415549403651165607299319, −2.04252804209525565536207341093, −1.99546764743876528837745447358, −1.57728332918893853765682201915, −1.41474294288857573765650224461, −0.45898129849462273206057108039, −0.20885354321755968706976410599,
0.20885354321755968706976410599, 0.45898129849462273206057108039, 1.41474294288857573765650224461, 1.57728332918893853765682201915, 1.99546764743876528837745447358, 2.04252804209525565536207341093, 2.37757415549403651165607299319, 2.82605179923579814619731568052, 2.84561326542897479118525896008, 3.00835853466204950898612402687, 3.74344345731746729832954448024, 3.95127181805860418111835907324, 3.97351412925969355109850596586, 4.02122885964939440890393802578, 4.62871576144711991545922893177, 4.74490439064127255540267277504, 4.76150642500062643971533485193, 5.03126809022214819815043317543, 5.17335353397650782263019465923, 5.56503337399670877144535030039, 5.99096607040410814367549485677, 6.13634146391230126780042335600, 6.33732903377607424362326405054, 6.48740459502075039145385782136, 6.57263687288255213501342879076
