| L(s) = 1 | + 53·9-s − 38·11-s + 182·19-s + 544·29-s − 460·31-s + 234·41-s + 650·49-s − 624·59-s + 340·61-s − 104·71-s − 2.10e3·79-s + 2.08e3·81-s − 1.59e3·89-s − 2.01e3·99-s + 972·101-s − 252·109-s − 1.57e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.25e3·169-s + ⋯ |
| L(s) = 1 | + 1.96·9-s − 1.04·11-s + 2.19·19-s + 3.48·29-s − 2.66·31-s + 0.891·41-s + 1.89·49-s − 1.37·59-s + 0.713·61-s − 0.173·71-s − 3.00·79-s + 2.85·81-s − 1.90·89-s − 2.04·99-s + 0.957·101-s − 0.221·109-s − 1.18·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.93·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.935133317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.935133317\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 53 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 650 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 19 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4250 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4201 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 91 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 5942 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 272 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 230 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 68182 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 117 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 20630 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 204942 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 136150 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 312 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 170 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 19357 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 52 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 184327 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1054 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1020373 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 799 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 899902 T^{2} + p^{6} T^{4} \) |
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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
−12.23389205751739045917647040635, −11.99099664498084366096494496948, −11.17379741080088000521611583393, −10.70839512519486331035953078984, −10.13344881585589638924205850822, −10.00742867766128240167464351816, −9.367714831589838942853812347706, −8.848087344515136778015542894947, −8.103018731224878711937240678754, −7.57310535014205494929287834914, −7.13261739251179517905256230814, −6.89490890536218954997138634845, −5.82532264956870446990620749060, −5.39785322726497494392861244111, −4.65384941255522121190455768147, −4.23616970281721007356707412749, −3.31134914687291845741414549016, −2.64511483351913043286358179974, −1.53802374098517992953125915134, −0.819866135306952500963460780247,
0.819866135306952500963460780247, 1.53802374098517992953125915134, 2.64511483351913043286358179974, 3.31134914687291845741414549016, 4.23616970281721007356707412749, 4.65384941255522121190455768147, 5.39785322726497494392861244111, 5.82532264956870446990620749060, 6.89490890536218954997138634845, 7.13261739251179517905256230814, 7.57310535014205494929287834914, 8.103018731224878711937240678754, 8.848087344515136778015542894947, 9.367714831589838942853812347706, 10.00742867766128240167464351816, 10.13344881585589638924205850822, 10.70839512519486331035953078984, 11.17379741080088000521611583393, 11.99099664498084366096494496948, 12.23389205751739045917647040635
