| 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
| \(L(2)\) |
\(\approx\) |
\(19.04941369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(19.04941369\) |
| \(L(\frac{5}{2})\) |
|
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
−10.94826705527549789951457968561, −10.83332764987046128352844403802, −9.760474019427137677442984441526, −9.686093146921244222301113634369, −8.784020859390785041390781401899, −8.645453166227106333881918707611, −8.227462043810885335529392441679, −8.048677763764230405898927565495, −7.19726739581177843532159752185, −7.18998965101606303987653175214, −6.62336266089388409498454841080, −5.99660369516687135390217843673, −4.93799221350388087564441150223, −4.43533566661621721496821003341, −3.84509133227563489834215382424, −3.17824105483652008526927068339, −2.50963361618824315808548965487, −2.37652648658096049057168116665, −1.87856241144005307199043095477, −1.30530853611715698700040911666,
1.30530853611715698700040911666, 1.87856241144005307199043095477, 2.37652648658096049057168116665, 2.50963361618824315808548965487, 3.17824105483652008526927068339, 3.84509133227563489834215382424, 4.43533566661621721496821003341, 4.93799221350388087564441150223, 5.99660369516687135390217843673, 6.62336266089388409498454841080, 7.18998965101606303987653175214, 7.19726739581177843532159752185, 8.048677763764230405898927565495, 8.227462043810885335529392441679, 8.645453166227106333881918707611, 8.784020859390785041390781401899, 9.686093146921244222301113634369, 9.760474019427137677442984441526, 10.83332764987046128352844403802, 10.94826705527549789951457968561
