| L(s) = 1 | − 8·3-s + 27·9-s − 74·25-s − 136·27-s − 680·31-s − 868·37-s + 1.37e3·49-s + 832·67-s + 592·75-s + 1.08e3·81-s + 5.44e3·93-s + 68·97-s − 4.68e3·103-s + 6.94e3·111-s − 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s − 1.09e4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.78e3·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 9-s − 0.591·25-s − 0.969·27-s − 3.93·31-s − 3.85·37-s + 4·49-s + 1.51·67-s + 0.911·75-s + 1.49·81-s + 6.06·93-s + 0.0711·97-s − 4.48·103-s + 5.93·111-s − 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 6.15·147-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 4·169-s + 0.000439·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1679352352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1679352352\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + 8 T + 37 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 199 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} )( 1 + 18 T + 199 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 108 T - 503 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} )( 1 + 108 T - 503 T^{2} + 108 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 340 T + 85809 T^{2} + 340 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 434 T + 137703 T^{2} + 434 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2}( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 738 T + p^{3} T^{2} )^{2}( 1 + 738 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 720 T + 313021 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} )( 1 + 720 T + 313021 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 416 T - 127707 T^{2} - 416 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 612 T + 16633 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} )( 1 + 612 T + 16633 T^{2} + 612 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 73 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 1674 T + 2097307 T^{2} - 1674 p^{3} T^{3} + p^{6} T^{4} )( 1 + 1674 T + 2097307 T^{2} + 1674 p^{3} T^{3} + p^{6} T^{4} ) \) |
| 97 | $C_2^2$ | \( ( 1 - 34 T - 911517 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.40857997492150532925498939323, −6.98333835160889542529262768033, −6.86194953369161030691005768824, −6.75247268377493265300800300050, −6.55628365082069979200652743241, −6.06930515550060915144682174023, −5.75381837180462287369536996275, −5.60154219534207920452032192685, −5.36937206770353546404721977701, −5.32439067396000519208958558051, −5.21743505059427081095138883941, −4.77317315973740750776589319256, −4.30204413061990723771605767100, −3.87481588122983611703504307194, −3.85836046963912228644096406726, −3.73304740175502106405738886968, −3.39863833377873652404014848042, −2.72739220450400653567297333197, −2.58524513078746341335265009706, −1.97712140167453155354317439823, −1.82493939414226015478476299289, −1.53735823961345379481586348967, −1.01564968400127826543567921057, −0.45309091328115380591277767427, −0.10921367093147978884609557250,
0.10921367093147978884609557250, 0.45309091328115380591277767427, 1.01564968400127826543567921057, 1.53735823961345379481586348967, 1.82493939414226015478476299289, 1.97712140167453155354317439823, 2.58524513078746341335265009706, 2.72739220450400653567297333197, 3.39863833377873652404014848042, 3.73304740175502106405738886968, 3.85836046963912228644096406726, 3.87481588122983611703504307194, 4.30204413061990723771605767100, 4.77317315973740750776589319256, 5.21743505059427081095138883941, 5.32439067396000519208958558051, 5.36937206770353546404721977701, 5.60154219534207920452032192685, 5.75381837180462287369536996275, 6.06930515550060915144682174023, 6.55628365082069979200652743241, 6.75247268377493265300800300050, 6.86194953369161030691005768824, 6.98333835160889542529262768033, 7.40857997492150532925498939323
