| L(s) = 1 | + 3.34e4·7-s + 2.90e5·9-s − 1.38e7·17-s − 3.72e7·23-s + 7.43e7·25-s − 1.05e8·31-s − 6.16e8·41-s + 5.37e9·47-s − 3.11e9·49-s + 9.73e9·63-s − 1.95e10·71-s − 2.92e9·73-s + 7.62e10·79-s + 5.31e10·81-s + 4.99e10·89-s + 1.50e11·97-s + 4.51e11·103-s − 1.70e11·113-s − 4.62e11·119-s + 2.84e11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4.01e12·153-s + ⋯ |
| L(s) = 1 | + 0.753·7-s + 1.64·9-s − 2.35·17-s − 1.20·23-s + 1.52·25-s − 0.663·31-s − 0.830·41-s + 3.41·47-s − 1.57·49-s + 1.23·63-s − 1.28·71-s − 0.165·73-s + 2.78·79-s + 1.69·81-s + 0.948·89-s + 1.77·97-s + 3.83·103-s − 0.869·113-s − 1.77·119-s + 0.998·121-s − 3.87·153-s − 0.909·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(4.828508127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.828508127\) |
| \(L(\frac{13}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 3590 p^{4} T^{2} + p^{22} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2973094 p^{2} T^{2} + p^{22} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2392 p T + p^{11} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 284813350678 T^{2} + p^{22} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3250539591430 T^{2} + p^{22} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6905934 T + p^{11} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 119314641380038 T^{2} + p^{22} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 18643272 T + p^{11} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 7912756903454758 T^{2} + p^{22} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 52843168 T + p^{11} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 322633551760058230 T^{2} + p^{22} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 308120442 T + p^{11} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1858294189067944150 T^{2} + p^{22} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2687348496 T + p^{11} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 15990678067626115990 T^{2} + p^{22} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 33383941434245697718 T^{2} + p^{22} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 38635239848010367078 T^{2} + p^{22} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4573302143790884710 T^{2} + p^{22} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9791485272 T + p^{11} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 1463791322 T + p^{11} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 38116845680 T + p^{11} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - \)\(17\!\cdots\!10\)\( T^{2} + p^{22} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 24992917110 T + p^{11} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 75013568546 T + p^{11} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.32394876768805268643001915499, −10.00377301217573836702793710041, −9.110651470307590233323603198794, −9.035918656374600845697955894721, −8.521439484605351261879212200072, −7.79033683178480988145693039390, −7.41975118486474722585633121860, −6.99565904177631937222669027263, −6.42331077006785696948417421072, −6.12288393029425129222884992473, −5.09617586295505686955744056456, −4.85337946903035835533733557232, −4.27481368932643670188059330801, −4.02216309695980623188817407932, −3.27673451371320770854529727683, −2.42238912355134045656027474980, −1.85703657132688955076378683499, −1.77606892777127041953409347500, −0.795404227159861550504585360639, −0.49747825090964987213239593180,
0.49747825090964987213239593180, 0.795404227159861550504585360639, 1.77606892777127041953409347500, 1.85703657132688955076378683499, 2.42238912355134045656027474980, 3.27673451371320770854529727683, 4.02216309695980623188817407932, 4.27481368932643670188059330801, 4.85337946903035835533733557232, 5.09617586295505686955744056456, 6.12288393029425129222884992473, 6.42331077006785696948417421072, 6.99565904177631937222669027263, 7.41975118486474722585633121860, 7.79033683178480988145693039390, 8.521439484605351261879212200072, 9.035918656374600845697955894721, 9.110651470307590233323603198794, 10.00377301217573836702793710041, 10.32394876768805268643001915499
