| L(s) = 1 | − 4·4-s + 29·9-s − 2·11-s + 16·16-s − 60·19-s − 158·29-s − 424·31-s − 116·36-s − 616·41-s + 8·44-s − 49·49-s − 1.25e3·59-s − 1.36e3·61-s − 64·64-s + 1.48e3·71-s + 240·76-s + 814·79-s + 112·81-s + 1.76e3·89-s − 58·99-s + 108·101-s + 3.22e3·109-s + 632·116-s − 2.65e3·121-s + 1.69e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.07·9-s − 0.0548·11-s + 1/4·16-s − 0.724·19-s − 1.01·29-s − 2.45·31-s − 0.537·36-s − 2.34·41-s + 0.0274·44-s − 1/7·49-s − 2.77·59-s − 2.87·61-s − 1/8·64-s + 2.48·71-s + 0.362·76-s + 1.15·79-s + 0.153·81-s + 2.09·89-s − 0.0588·99-s + 0.106·101-s + 2.83·109-s + 0.505·116-s − 1.99·121-s + 1.22·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.114262427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.114262427\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 29 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4345 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 p^{2} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21834 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 79 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 212 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65206 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 308 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 19070 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 193005 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 143142 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 628 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 684 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 513610 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 744 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 250958 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 407 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 728838 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 880 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{6} T^{4} \) |
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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
−11.07270180442815969321628227944, −10.74633803622088281574566360625, −10.54485834712460234460787477678, −9.677543605440732413888885912170, −9.593651704772237789296110441062, −8.950417078005054983916331323567, −8.676168647261761542792114200288, −7.84859590328675160557527372360, −7.55539685193268587485368507273, −7.13816393922341015604990907967, −6.36200864139254039455650288370, −6.10486090019485429028840254377, −5.11662555270386357107861196385, −4.98208747630094061452272929354, −4.21951863896336391526921547425, −3.63852137078612013256122339294, −3.20250392181084573141852854969, −1.86529752470141688044402175967, −1.69977290143386487368605479024, −0.36824639769648774490223310114,
0.36824639769648774490223310114, 1.69977290143386487368605479024, 1.86529752470141688044402175967, 3.20250392181084573141852854969, 3.63852137078612013256122339294, 4.21951863896336391526921547425, 4.98208747630094061452272929354, 5.11662555270386357107861196385, 6.10486090019485429028840254377, 6.36200864139254039455650288370, 7.13816393922341015604990907967, 7.55539685193268587485368507273, 7.84859590328675160557527372360, 8.676168647261761542792114200288, 8.950417078005054983916331323567, 9.593651704772237789296110441062, 9.677543605440732413888885912170, 10.54485834712460234460787477678, 10.74633803622088281574566360625, 11.07270180442815969321628227944
