| L(s) = 1 | − 2-s + 39·4-s + 112·7-s − 109·8-s − 248·11-s − 876·13-s − 112·14-s + 599·16-s + 2.03e3·17-s + 1.46e3·19-s + 248·22-s − 3.21e3·23-s + 876·26-s + 4.36e3·28-s − 1.94e3·29-s + 2.67e3·31-s − 4.96e3·32-s − 2.03e3·34-s − 8.66e3·37-s − 1.46e3·38-s + 7.62e3·41-s + 1.64e4·43-s − 9.67e3·44-s + 3.21e3·46-s − 1.93e4·47-s + 1.97e3·49-s − 3.41e4·52-s + ⋯ |
| L(s) = 1 | − 0.176·2-s + 1.21·4-s + 0.863·7-s − 0.602·8-s − 0.617·11-s − 1.43·13-s − 0.152·14-s + 0.584·16-s + 1.70·17-s + 0.930·19-s + 0.109·22-s − 1.26·23-s + 0.254·26-s + 1.05·28-s − 0.430·29-s + 0.499·31-s − 0.857·32-s − 0.302·34-s − 1.04·37-s − 0.164·38-s + 0.708·41-s + 1.35·43-s − 0.753·44-s + 0.224·46-s − 1.27·47-s + 0.117·49-s − 1.75·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.794498676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.794498676\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T - 19 p T^{2} + p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 16 p T + 10574 T^{2} - 16 p^{6} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 248 T + 232774 T^{2} + 248 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 876 T + 908254 T^{2} + 876 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2036 T + 3221638 T^{2} - 2036 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1464 T + 1064278 T^{2} - 1464 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3216 T + 15222766 T^{2} + 3216 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1948 T + 41552158 T^{2} + 1948 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2672 T + 55875902 T^{2} - 2672 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8668 T + 89387694 T^{2} + 8668 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7628 T + 215999542 T^{2} - 7628 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 16440 T + 361166470 T^{2} - 16440 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 19360 T + 397194910 T^{2} + 19360 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 14356 T + 360101806 T^{2} + 14356 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 904 T + 361967398 T^{2} - 904 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20220 T + 1540010398 T^{2} - 20220 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12904 T + 2574352118 T^{2} - 12904 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 40976 T + 2981176846 T^{2} - 40976 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 59124 T + 4982256886 T^{2} + 59124 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 107600 T + 7602983198 T^{2} - 107600 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 122088 T + 9942170518 T^{2} + 122088 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 103764 T + 8272750678 T^{2} + 103764 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24764 T + 1402516038 T^{2} - 24764 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.80428459109377025919879648316, −10.94711835653332622932186157275, −10.89654072699237975700792579357, −10.13637614200352679576046685532, −9.579574638686533379952916843824, −9.570042433107887483698505745972, −8.414950744523001186693011337933, −7.994002174123247852276822130745, −7.66693729395527402766221496466, −7.18387966472596569509440027198, −6.71993197012630274389811715862, −5.78372282197290060449300210464, −5.55808162907645300226459163858, −4.96527449902666003914136139910, −4.17047232413730906313045465441, −3.23444094564056078794299897299, −2.75614155832381568605168158463, −2.05344153657873011244517482183, −1.47605910012614867788937766849, −0.48480607793043219440562529010,
0.48480607793043219440562529010, 1.47605910012614867788937766849, 2.05344153657873011244517482183, 2.75614155832381568605168158463, 3.23444094564056078794299897299, 4.17047232413730906313045465441, 4.96527449902666003914136139910, 5.55808162907645300226459163858, 5.78372282197290060449300210464, 6.71993197012630274389811715862, 7.18387966472596569509440027198, 7.66693729395527402766221496466, 7.994002174123247852276822130745, 8.414950744523001186693011337933, 9.570042433107887483698505745972, 9.579574638686533379952916843824, 10.13637614200352679576046685532, 10.89654072699237975700792579357, 10.94711835653332622932186157275, 11.80428459109377025919879648316
