| L(s) = 1 | − 4-s − 2·5-s − 9-s − 10·11-s + 16-s + 10·19-s + 2·20-s − 25-s + 4·29-s + 2·31-s + 36-s + 12·41-s + 10·44-s + 2·45-s + 13·49-s + 20·55-s − 20·59-s − 28·61-s − 64-s − 18·71-s − 10·76-s − 10·79-s − 2·80-s + 81-s − 6·89-s − 20·95-s + 10·99-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s − 1/3·9-s − 3.01·11-s + 1/4·16-s + 2.29·19-s + 0.447·20-s − 1/5·25-s + 0.742·29-s + 0.359·31-s + 1/6·36-s + 1.87·41-s + 1.50·44-s + 0.298·45-s + 13/7·49-s + 2.69·55-s − 2.60·59-s − 3.58·61-s − 1/8·64-s − 2.13·71-s − 1.14·76-s − 1.12·79-s − 0.223·80-s + 1/9·81-s − 0.635·89-s − 2.05·95-s + 1.00·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5467039986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5467039986\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−10.18874160642155211997753089142, −10.10600486064744863422335040363, −9.211098191991994807640338601733, −9.192122096228278683362482097397, −8.561884869855235754382523425160, −7.953132079078275551480790536103, −7.64378437401921216055901242366, −7.61616000176016779642244096715, −7.29121855227043982553249199146, −6.22374225769867074162870602737, −5.87805932560752718461395752671, −5.30221655571935217398086244202, −5.19178200871947861699615301855, −4.36839168767893088398497180253, −4.29577267265008237659604437346, −3.12450224960058915117254787891, −2.98857294562736104201925937535, −2.63243592550045901050677890798, −1.40172750430398820767725177859, −0.36339773528994289038468446489,
0.36339773528994289038468446489, 1.40172750430398820767725177859, 2.63243592550045901050677890798, 2.98857294562736104201925937535, 3.12450224960058915117254787891, 4.29577267265008237659604437346, 4.36839168767893088398497180253, 5.19178200871947861699615301855, 5.30221655571935217398086244202, 5.87805932560752718461395752671, 6.22374225769867074162870602737, 7.29121855227043982553249199146, 7.61616000176016779642244096715, 7.64378437401921216055901242366, 7.953132079078275551480790536103, 8.561884869855235754382523425160, 9.192122096228278683362482097397, 9.211098191991994807640338601733, 10.10600486064744863422335040363, 10.18874160642155211997753089142
