| L(s) = 1 | + 2·2-s − 12·3-s − 12·4-s − 24·6-s − 56·8-s − 54·9-s − 52·11-s + 144·12-s + 80·16-s − 452·17-s − 108·18-s + 268·19-s − 104·22-s + 672·24-s + 2.05e3·27-s + 1.05e3·32-s + 624·33-s − 904·34-s + 648·36-s + 536·38-s + 1.98e3·41-s + 3.76e3·43-s + 624·44-s − 960·48-s + 962·49-s + 5.42e3·51-s + 4.10e3·54-s + ⋯ |
| L(s) = 1 | + 1/2·2-s − 4/3·3-s − 3/4·4-s − 2/3·6-s − 7/8·8-s − 2/3·9-s − 0.429·11-s + 12-s + 5/16·16-s − 1.56·17-s − 1/3·18-s + 0.742·19-s − 0.214·22-s + 7/6·24-s + 2.81·27-s + 1.03·32-s + 0.573·33-s − 0.782·34-s + 1/2·36-s + 0.371·38-s + 1.18·41-s + 2.03·43-s + 0.322·44-s − 0.416·48-s + 0.400·49-s + 2.08·51-s + 1.40·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.06977740999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06977740999\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 - p T + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + 2 p T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 962 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 26 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 56162 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 226 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 134 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 463682 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1298402 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 311042 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 629282 T^{2} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 994 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 1882 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5320322 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1257122 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5018 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 23382242 T^{2} + p^{8} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8006 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 50512322 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 386 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 43766398 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2234 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10046 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8738 T + p^{4} T^{2} )^{2} \) |
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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
−11.91007867327574786101469805756, −11.73964667817038333647268503980, −10.96002512317125532442430936938, −10.82980382766473498739801789591, −10.36476646109368584225908938261, −9.269131132794689480954226548109, −9.239037110453138290096941136876, −8.675457832443711883737116760594, −7.970548299690277585854070114724, −7.44080020505757299548467807323, −6.54858204186049875498840505906, −6.12330463596332988675394017146, −5.62791371753667337712793619102, −5.31292342739507914770077987271, −4.45727213600778096230686963957, −4.28431344195366768788410073752, −2.97955347959589371645050932179, −2.68989705148598959624813232157, −1.15262362326389728550929966209, −0.10696605963580434954571157664,
0.10696605963580434954571157664, 1.15262362326389728550929966209, 2.68989705148598959624813232157, 2.97955347959589371645050932179, 4.28431344195366768788410073752, 4.45727213600778096230686963957, 5.31292342739507914770077987271, 5.62791371753667337712793619102, 6.12330463596332988675394017146, 6.54858204186049875498840505906, 7.44080020505757299548467807323, 7.970548299690277585854070114724, 8.675457832443711883737116760594, 9.239037110453138290096941136876, 9.269131132794689480954226548109, 10.36476646109368584225908938261, 10.82980382766473498739801789591, 10.96002512317125532442430936938, 11.73964667817038333647268503980, 11.91007867327574786101469805756
