| L(s) = 1 | + 25·7-s + 427·13-s − 3.42e3·19-s + 3.12e3·25-s + 1.03e4·31-s − 1.33e4·37-s + 3.35e3·43-s + 1.68e4·49-s − 5.69e4·61-s + 3.79e4·67-s + 1.59e5·73-s − 9.08e4·79-s + 1.06e4·91-s − 1.77e5·97-s + 2.11e5·103-s + 2.28e5·109-s + 1.61e5·121-s + 127-s + 131-s − 8.55e4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.192·7-s + 0.700·13-s − 2.17·19-s + 25-s + 1.92·31-s − 1.59·37-s + 0.276·43-s + 49-s − 1.95·61-s + 1.03·67-s + 3.49·73-s − 1.63·79-s + 0.135·91-s − 1.91·97-s + 1.96·103-s + 1.84·109-s + 121-s + 5.50e−6·127-s + 5.09e−6·131-s − 0.419·133-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.423797943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.423797943\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 236 T + p^{5} T^{2} )( 1 + 211 T + p^{5} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 1202 T + p^{5} T^{2} )( 1 + 775 T + p^{5} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 1711 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7601 T + p^{5} T^{2} )( 1 - 2723 T + p^{5} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6661 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 22475 T + p^{5} T^{2} )( 1 + 19123 T + p^{5} T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 18301 T + p^{5} T^{2} )( 1 + 38626 T + p^{5} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 73475 T + p^{5} T^{2} )( 1 + 35536 T + p^{5} T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 79577 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 9707 T + p^{5} T^{2} )( 1 + 100564 T + p^{5} T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 43339 T + p^{5} T^{2} )( 1 + 134386 T + p^{5} T^{2} ) \) |
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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
−10.85706466799096712258588530963, −10.64286901632863572827981489191, −10.22429634096267184470409334166, −9.651443594097396467158625701308, −8.967573361605869797293268386286, −8.633913285538524613744739544284, −8.309968612355185030363962518428, −7.81845227692924545902685406221, −7.02840134354759763603671829339, −6.63004505118201672768426410554, −6.24642187158265625120141043419, −5.66753491988871419698198422914, −4.91143843556486445278362179279, −4.51222937171788117549914614481, −3.92522197718206639762401838817, −3.27822394577811325247820730646, −2.52302441967619260528726252681, −1.95676388904515915067010982105, −1.14884732042418401876045070323, −0.44345272778426270379010219929,
0.44345272778426270379010219929, 1.14884732042418401876045070323, 1.95676388904515915067010982105, 2.52302441967619260528726252681, 3.27822394577811325247820730646, 3.92522197718206639762401838817, 4.51222937171788117549914614481, 4.91143843556486445278362179279, 5.66753491988871419698198422914, 6.24642187158265625120141043419, 6.63004505118201672768426410554, 7.02840134354759763603671829339, 7.81845227692924545902685406221, 8.309968612355185030363962518428, 8.633913285538524613744739544284, 8.967573361605869797293268386286, 9.651443594097396467158625701308, 10.22429634096267184470409334166, 10.64286901632863572827981489191, 10.85706466799096712258588530963
