| L(s) = 1 | − 4·4-s − 10·5-s − 9·9-s + 104·11-s + 16·16-s + 38·19-s + 40·20-s − 25·25-s + 620·29-s − 456·31-s + 36·36-s − 756·41-s − 416·44-s + 90·45-s + 650·49-s − 1.04e3·55-s + 640·59-s + 564·61-s − 64·64-s − 936·71-s − 152·76-s − 1.64e3·79-s − 160·80-s + 81·81-s − 1.90e3·89-s − 380·95-s − 936·99-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 1/3·9-s + 2.85·11-s + 1/4·16-s + 0.458·19-s + 0.447·20-s − 1/5·25-s + 3.97·29-s − 2.64·31-s + 1/6·36-s − 2.87·41-s − 1.42·44-s + 0.298·45-s + 1.89·49-s − 2.54·55-s + 1.41·59-s + 1.18·61-s − 1/8·64-s − 1.56·71-s − 0.229·76-s − 2.33·79-s − 0.223·80-s + 1/9·81-s − 2.26·89-s − 0.410·95-s − 0.950·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.193131652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.193131652\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 p T + p^{3} T^{2} \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 650 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 52 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1030 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2770 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 24010 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 310 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 228 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 26230 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 378 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 42050 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 152890 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 65430 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 320 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 282 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 367270 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 468 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 573730 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 820 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1104370 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 950 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 584350 T^{2} + p^{6} T^{4} \) |
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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
−10.35969235422825923045216564786, −10.06868765215357352483447169548, −9.793625280760153326937139358495, −8.881913300019563080299903614893, −8.800705971482420419519519834734, −8.634899472880471196207106808945, −8.048594117723330106706983672991, −7.23751245950678363242784978683, −6.83251251932031609010438492329, −6.81094115302663656408561991147, −5.88800250900887038217364202060, −5.61797119473055241538155062209, −4.66337700327224600264450738549, −4.48052090957586706263830010178, −3.66447685257346541481968161914, −3.66082992080344898277200583515, −2.87579269519403606571054032500, −1.78782135397144817205593003202, −1.16840470289457850181689430594, −0.51949839166847472751029869514,
0.51949839166847472751029869514, 1.16840470289457850181689430594, 1.78782135397144817205593003202, 2.87579269519403606571054032500, 3.66082992080344898277200583515, 3.66447685257346541481968161914, 4.48052090957586706263830010178, 4.66337700327224600264450738549, 5.61797119473055241538155062209, 5.88800250900887038217364202060, 6.81094115302663656408561991147, 6.83251251932031609010438492329, 7.23751245950678363242784978683, 8.048594117723330106706983672991, 8.634899472880471196207106808945, 8.800705971482420419519519834734, 8.881913300019563080299903614893, 9.793625280760153326937139358495, 10.06868765215357352483447169548, 10.35969235422825923045216564786
