| L(s) = 1 | + 2·5-s + 2·11-s + 6·13-s − 8·17-s − 19-s + 8·23-s + 5·25-s − 8·29-s + 3·31-s + 37-s + 12·41-s + 22·43-s − 6·47-s − 12·53-s + 4·55-s − 4·59-s − 6·61-s + 12·65-s − 13·67-s + 20·71-s − 11·73-s + 3·79-s + 4·83-s − 16·85-s − 2·95-s − 20·97-s − 10·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.603·11-s + 1.66·13-s − 1.94·17-s − 0.229·19-s + 1.66·23-s + 25-s − 1.48·29-s + 0.538·31-s + 0.164·37-s + 1.87·41-s + 3.35·43-s − 0.875·47-s − 1.64·53-s + 0.539·55-s − 0.520·59-s − 0.768·61-s + 1.48·65-s − 1.58·67-s + 2.37·71-s − 1.28·73-s + 0.337·79-s + 0.439·83-s − 1.73·85-s − 0.205·95-s − 2.03·97-s − 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.388522001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.388522001\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.340101777296431115495343362653, −9.175550666949976444772416203353, −8.765675855405502971298834920763, −8.598943728540689125764411132755, −7.87282857043256791873736466077, −7.53212569949940837434048161319, −6.95911482504656434009317315854, −6.73530093561808655321780487484, −6.13676119711765140903536923150, −5.94437176829392206325765159045, −5.74142686815084228301841584956, −4.88303167738140100368889216539, −4.43649725074224532532011486599, −4.30848011227964086679683880672, −3.55133581738399592637419895008, −3.08688043713126308262291844766, −2.49995907952739034186634807640, −1.98297348320389184406725278706, −1.34866671483083759308299642893, −0.74784149754909917641218120717,
0.74784149754909917641218120717, 1.34866671483083759308299642893, 1.98297348320389184406725278706, 2.49995907952739034186634807640, 3.08688043713126308262291844766, 3.55133581738399592637419895008, 4.30848011227964086679683880672, 4.43649725074224532532011486599, 4.88303167738140100368889216539, 5.74142686815084228301841584956, 5.94437176829392206325765159045, 6.13676119711765140903536923150, 6.73530093561808655321780487484, 6.95911482504656434009317315854, 7.53212569949940837434048161319, 7.87282857043256791873736466077, 8.598943728540689125764411132755, 8.765675855405502971298834920763, 9.175550666949976444772416203353, 9.340101777296431115495343362653
