| L(s) = 1 | − 64·7-s − 10·9-s − 196·17-s + 64·23-s + 106·25-s − 512·31-s − 204·41-s − 640·47-s + 2.38e3·49-s + 640·63-s − 832·71-s − 276·73-s − 128·79-s − 629·81-s + 1.16e3·89-s + 476·97-s + 1.98e3·103-s − 604·113-s + 1.25e4·119-s + 2.59e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.96e3·153-s + ⋯ |
| L(s) = 1 | − 3.45·7-s − 0.370·9-s − 2.79·17-s + 0.580·23-s + 0.847·25-s − 2.96·31-s − 0.777·41-s − 1.98·47-s + 6.95·49-s + 1.27·63-s − 1.39·71-s − 0.442·73-s − 0.182·79-s − 0.862·81-s + 1.38·89-s + 0.498·97-s + 1.89·103-s − 0.502·113-s + 9.66·119-s + 1.95·121-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 + 1.03·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 106 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2598 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 98 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 5974 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 19194 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 256 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 92842 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 102 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 71398 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 320 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 291978 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 244294 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 49466 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 296822 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 416 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 138 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 64 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 989910 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 582 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 238 T + p^{3} T^{2} )^{2} \) |
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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
−12.78634401478974716182020570086, −12.45092757056855816104567684050, −11.37923953594496628618316219292, −11.20949164116986533597062722664, −10.21450542201675788116760123957, −10.20760603513653161292808924193, −9.212338049798465062086091506886, −8.953014507280457733055146880209, −8.903006379956564090130312162996, −7.48396867514535230660036209520, −6.82132329402451852400860869549, −6.63439182096268572137556698852, −6.15864662847546624487554837769, −5.36702368536121864314502922349, −4.35786704626479886346482269382, −3.46338842835132231093544429303, −3.12876406372654968398368489262, −2.20797048657158391561318135753, 0, 0,
2.20797048657158391561318135753, 3.12876406372654968398368489262, 3.46338842835132231093544429303, 4.35786704626479886346482269382, 5.36702368536121864314502922349, 6.15864662847546624487554837769, 6.63439182096268572137556698852, 6.82132329402451852400860869549, 7.48396867514535230660036209520, 8.903006379956564090130312162996, 8.953014507280457733055146880209, 9.212338049798465062086091506886, 10.20760603513653161292808924193, 10.21450542201675788116760123957, 11.20949164116986533597062722664, 11.37923953594496628618316219292, 12.45092757056855816104567684050, 12.78634401478974716182020570086
