| L(s) = 1 | − 40·7-s + 50·9-s − 132·17-s − 344·23-s + 214·25-s − 256·31-s − 404·41-s − 816·47-s + 514·49-s − 2.00e3·63-s + 1.40e3·71-s + 836·73-s + 1.48e3·79-s + 1.77e3·81-s + 164·89-s − 2.24e3·97-s − 1.57e3·103-s − 1.08e3·113-s + 5.28e3·119-s + 2.46e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 6.60e3·153-s + ⋯ |
| L(s) = 1 | − 2.15·7-s + 1.85·9-s − 1.88·17-s − 3.11·23-s + 1.71·25-s − 1.48·31-s − 1.53·41-s − 2.53·47-s + 1.49·49-s − 3.99·63-s + 2.34·71-s + 1.34·73-s + 2.11·79-s + 2.42·81-s + 0.195·89-s − 2.34·97-s − 1.50·103-s − 0.902·113-s + 4.06·119-s + 1.85·121-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 − 3.48·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5986117787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5986117787\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 214 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2466 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1478 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 66 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 12526 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 172 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 48774 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 128 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 76342 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 202 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70210 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 408 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 178346 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 307074 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 365158 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 560722 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 700 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 418 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 744 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 683890 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 82 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1122 T + p^{3} T^{2} )^{2} \) |
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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
−12.12532635346900776171116018931, −11.13766477335058199241036197796, −10.84838661678314578719945356647, −10.20933697575901816904992928190, −9.854070737280347293667452203539, −9.478500063167690374169938011415, −9.275373929671376793805333079964, −8.180534893498519379782528293148, −8.143870539090073500484143449758, −6.92050518766229709018673813712, −6.82852509645970742603065821296, −6.57501172101635111372119675474, −5.94126336802432237634777685809, −4.98543775143759948283876500824, −4.45486216720036983971795561980, −3.62195825889249314631442671288, −3.53848057744511487794138093367, −2.29322460015844818711690119812, −1.72540361878266596083152973375, −0.29291374160898331249122885786,
0.29291374160898331249122885786, 1.72540361878266596083152973375, 2.29322460015844818711690119812, 3.53848057744511487794138093367, 3.62195825889249314631442671288, 4.45486216720036983971795561980, 4.98543775143759948283876500824, 5.94126336802432237634777685809, 6.57501172101635111372119675474, 6.82852509645970742603065821296, 6.92050518766229709018673813712, 8.143870539090073500484143449758, 8.180534893498519379782528293148, 9.275373929671376793805333079964, 9.478500063167690374169938011415, 9.854070737280347293667452203539, 10.20933697575901816904992928190, 10.84838661678314578719945356647, 11.13766477335058199241036197796, 12.12532635346900776171116018931
