| L(s) = 1 | − 4·4-s + 53·9-s − 64·11-s + 16·16-s − 72·19-s + 54·29-s − 66·31-s − 212·36-s − 314·41-s + 256·44-s + 362·49-s + 1.48e3·59-s + 1.10e3·61-s − 64·64-s + 1.39e3·71-s + 288·76-s + 1.28e3·79-s + 2.08e3·81-s + 204·89-s − 3.39e3·99-s − 12·101-s − 500·109-s − 216·116-s + 410·121-s + 264·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.96·9-s − 1.75·11-s + 1/4·16-s − 0.869·19-s + 0.345·29-s − 0.382·31-s − 0.981·36-s − 1.19·41-s + 0.877·44-s + 1.05·49-s + 3.28·59-s + 2.31·61-s − 1/8·64-s + 2.33·71-s + 0.434·76-s + 1.83·79-s + 2.85·81-s + 0.242·89-s − 3.44·99-s − 0.0118·101-s − 0.439·109-s − 0.172·116-s + 0.308·121-s + 0.191·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1322500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1322500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.078819259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.078819259\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 53 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 362 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2185 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9426 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 27 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 33 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 98170 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 157 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 158690 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 203421 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 297558 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 744 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 552 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 577190 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 699 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 407153 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 644 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 881430 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 102 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1491262 T^{2} + p^{6} T^{4} \) |
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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
−9.611623315354084441004582073718, −9.453268268429156239021701226952, −8.546120481622535361352528206214, −8.545185064752985026744878051610, −7.86789926941610879084074293664, −7.76759934325251520322990066354, −7.02980907065425941997839378711, −6.79279338172748989328886110233, −6.52553877139369898041758517807, −5.56384174029250106696254186452, −5.30070672834448934746670885091, −5.01239495065470418242703921624, −4.40771840936118259395765769420, −3.89754864107416668987693964554, −3.70700424576106211360027109790, −2.83914000070068142134147479566, −2.07391936084045200963661703409, −2.00473943284263327042443923987, −0.827849945342552178613483849407, −0.57110409237879660682512388797,
0.57110409237879660682512388797, 0.827849945342552178613483849407, 2.00473943284263327042443923987, 2.07391936084045200963661703409, 2.83914000070068142134147479566, 3.70700424576106211360027109790, 3.89754864107416668987693964554, 4.40771840936118259395765769420, 5.01239495065470418242703921624, 5.30070672834448934746670885091, 5.56384174029250106696254186452, 6.52553877139369898041758517807, 6.79279338172748989328886110233, 7.02980907065425941997839378711, 7.76759934325251520322990066354, 7.86789926941610879084074293664, 8.545185064752985026744878051610, 8.546120481622535361352528206214, 9.453268268429156239021701226952, 9.611623315354084441004582073718
