| L(s) = 1 | − 16·3-s + 15·4-s − 28·7-s + 138·9-s − 240·12-s + 161·16-s + 136·17-s − 160·19-s + 448·21-s − 236·23-s − 752·27-s − 420·28-s + 2.07e3·36-s − 268·37-s − 2.57e3·48-s − 98·49-s − 2.17e3·51-s + 2.56e3·57-s + 360·59-s − 3.86e3·63-s + 1.45e3·64-s + 2.04e3·68-s + 3.77e3·69-s − 1.65e3·73-s − 2.40e3·76-s + 1.93e3·81-s + 6.72e3·84-s + ⋯ |
| L(s) = 1 | − 3.07·3-s + 15/8·4-s − 1.51·7-s + 46/9·9-s − 5.77·12-s + 2.51·16-s + 1.94·17-s − 1.93·19-s + 4.65·21-s − 2.13·23-s − 5.36·27-s − 2.83·28-s + 9.58·36-s − 1.19·37-s − 7.74·48-s − 2/7·49-s − 5.97·51-s + 5.94·57-s + 0.794·59-s − 7.72·63-s + 2.84·64-s + 3.63·68-s + 6.58·69-s − 2.65·73-s − 3.62·76-s + 2.64·81-s + 8.72·84-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1399548761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1399548761\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 8 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{6} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2262 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 80 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 118 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 32902 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 54682 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 134 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 127842 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 85030 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 7650 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 114410 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 180 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 441862 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 252250 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 707722 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 828 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 793478 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 838870 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1490 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1376 T + p^{3} T^{2} )^{2} \) |
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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
−10.98331427895250149213930783382, −10.61309647394231599820996199012, −10.29426239868705078187495514854, −9.956245242154209799435762412300, −9.836178569640224529499373716024, −8.593043546861774463613604128801, −8.026934644918999633001956108029, −7.28381864162805249318513822867, −7.03725385481933540366396049317, −6.49523364766233869688273100759, −6.12264425848617884042279752055, −6.01124231391813385895536605858, −5.63334233584287910248058534642, −5.02444693701406433432415833681, −4.16510857777031907530480067285, −3.56259258782748810347260999421, −2.80076432898675624744932301579, −1.83971311097961428864213452523, −1.19789397534352385910691848825, −0.15722913712257133124419387649,
0.15722913712257133124419387649, 1.19789397534352385910691848825, 1.83971311097961428864213452523, 2.80076432898675624744932301579, 3.56259258782748810347260999421, 4.16510857777031907530480067285, 5.02444693701406433432415833681, 5.63334233584287910248058534642, 6.01124231391813385895536605858, 6.12264425848617884042279752055, 6.49523364766233869688273100759, 7.03725385481933540366396049317, 7.28381864162805249318513822867, 8.026934644918999633001956108029, 8.593043546861774463613604128801, 9.836178569640224529499373716024, 9.956245242154209799435762412300, 10.29426239868705078187495514854, 10.61309647394231599820996199012, 10.98331427895250149213930783382
