| L(s) = 1 | − 576·19-s − 400·25-s + 220·49-s + 2.88e3·67-s + 2.24e3·73-s + 800·97-s + 3.02e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8.14e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 6.95·19-s − 3.19·25-s + 0.641·49-s + 5.25·67-s + 3.59·73-s + 0.837·97-s + 2.26·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.70·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + 0.000300·223-s + 0.000292·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2152432177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2152432177\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 + 8 p^{2} T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 1510 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 92 T + p^{3} T^{2} )^{2}( 1 + 92 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 5776 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 144 T + p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 13966 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 34328 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 45182 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 100010 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 124720 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 51554 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 296696 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 244870 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 356038 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 720 T + p^{3} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 456622 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 560 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 856478 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 423574 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 449440 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 200 T + p^{3} T^{2} )^{4} \) |
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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
−6.56281034400439233085556041540, −6.43843192365836385820435054902, −6.29919462087536590828771356906, −6.00798359895126077386856891297, −5.77736440171137696972742726129, −5.68171788238998973914771774336, −5.05985750212497514351269240591, −4.97465695820976908967198997633, −4.95513503233515095063957414186, −4.36093176798429924482095722157, −4.26099774609517446424468682222, −3.97124111841621121185482421829, −3.88543018155538391758248696588, −3.78862602835156757988029135161, −3.52700506309597052308804334812, −3.00497551171020246463341863193, −2.35463435717504689591773928295, −2.30794370512606658577814912246, −2.16875559220079778456627779027, −2.04511821805550432782441263438, −1.96003129788142464707272072923, −1.23845866807430376956930803850, −0.825920818915050427934286309151, −0.34856632871782120269253601739, −0.094425600178660463685310290735,
0.094425600178660463685310290735, 0.34856632871782120269253601739, 0.825920818915050427934286309151, 1.23845866807430376956930803850, 1.96003129788142464707272072923, 2.04511821805550432782441263438, 2.16875559220079778456627779027, 2.30794370512606658577814912246, 2.35463435717504689591773928295, 3.00497551171020246463341863193, 3.52700506309597052308804334812, 3.78862602835156757988029135161, 3.88543018155538391758248696588, 3.97124111841621121185482421829, 4.26099774609517446424468682222, 4.36093176798429924482095722157, 4.95513503233515095063957414186, 4.97465695820976908967198997633, 5.05985750212497514351269240591, 5.68171788238998973914771774336, 5.77736440171137696972742726129, 6.00798359895126077386856891297, 6.29919462087536590828771356906, 6.43843192365836385820435054902, 6.56281034400439233085556041540
