| L(s) = 1 | + 12·7-s + 24·17-s − 12·23-s + 14·25-s + 24·47-s + 68·49-s − 12·71-s + 20·73-s − 36·79-s − 24·89-s + 16·97-s − 24·103-s + 24·113-s + 288·119-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 144·161-s + 163-s + 167-s + 34·169-s + 173-s + ⋯ |
| L(s) = 1 | + 4.53·7-s + 5.82·17-s − 2.50·23-s + 14/5·25-s + 3.50·47-s + 68/7·49-s − 1.42·71-s + 2.34·73-s − 4.05·79-s − 2.54·89-s + 1.62·97-s − 2.36·103-s + 2.25·113-s + 26.4·119-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 − 11.3·161-s + 0.0783·163-s + 0.0773·167-s + 2.61·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(27.62215182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.62215182\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 12 T + 67 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 714 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 38 T^{2} + 1611 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 106 T^{2} + 5115 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6954 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 810 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 10134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 202 T^{2} + 17211 T^{4} - 202 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 20106 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 79 | $D_{4}$ | \( ( 1 + 18 T + 212 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 187 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
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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
−5.70796987561597413363761177316, −5.49812411045069714931575740782, −5.44897349970160384710304239015, −5.33891028309593595284997172106, −5.12623979800253177214779860042, −4.67470552623576787282584452946, −4.63838765141743865266714465281, −4.50374931893853714082458011941, −4.36521242925281277582450480006, −4.09982045233401948721037458130, −3.98674286776720340377637047804, −3.47636940910132793093894671484, −3.42063568201087327618523670994, −3.17978656405344119400592734463, −3.08666090819395856337335943795, −2.60978414748067023648731594908, −2.52067518244370899344487873131, −2.18224472448400216683822737768, −1.78123070163247051046856331641, −1.77064894237902568981103887491, −1.41020536435015566580306835029, −1.25695659249976452179835682992, −1.01168657861581746451372445020, −0.952772847692637523358534820204, −0.54815504132620902722665549998,
0.54815504132620902722665549998, 0.952772847692637523358534820204, 1.01168657861581746451372445020, 1.25695659249976452179835682992, 1.41020536435015566580306835029, 1.77064894237902568981103887491, 1.78123070163247051046856331641, 2.18224472448400216683822737768, 2.52067518244370899344487873131, 2.60978414748067023648731594908, 3.08666090819395856337335943795, 3.17978656405344119400592734463, 3.42063568201087327618523670994, 3.47636940910132793093894671484, 3.98674286776720340377637047804, 4.09982045233401948721037458130, 4.36521242925281277582450480006, 4.50374931893853714082458011941, 4.63838765141743865266714465281, 4.67470552623576787282584452946, 5.12623979800253177214779860042, 5.33891028309593595284997172106, 5.44897349970160384710304239015, 5.49812411045069714931575740782, 5.70796987561597413363761177316
