| L(s) = 1 | + 2·2-s + 3·3-s − 5·4-s + 12·5-s + 6·6-s − 7·7-s − 20·8-s + 6·9-s + 24·10-s − 16·11-s − 15·12-s − 13·13-s − 14·14-s + 36·15-s + 5·16-s + 12·18-s + 39·19-s − 60·20-s − 21·21-s − 32·22-s + 8·23-s − 60·24-s + 71·25-s − 26·26-s + 9·27-s + 35·28-s + 38·29-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s − 5/4·4-s + 12/5·5-s + 6-s − 7-s − 5/2·8-s + 2/3·9-s + 12/5·10-s − 1.45·11-s − 5/4·12-s − 13-s − 14-s + 12/5·15-s + 5/16·16-s + 2/3·18-s + 2.05·19-s − 3·20-s − 21-s − 1.45·22-s + 8/23·23-s − 5/2·24-s + 2.83·25-s − 26-s + 1/3·27-s + 5/4·28-s + 1.31·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.949663611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.949663611\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 12 T + 73 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 16 T + 135 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 470 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 39 T + 868 T^{2} - 39 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 38 T + 603 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 36 T + 1393 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 72 T + 3409 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 29 T - 1008 T^{2} + 29 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T + 3181 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 40 T - 1209 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2630 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 121 T + p^{2} T^{2} )( 1 - 74 T + p^{2} T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 38 T - 3045 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 122 T + 9843 T^{2} - 122 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 93 T + 8212 T^{2} - 93 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 110 T + 5859 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13586 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 5326 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 153 T + 17212 T^{2} + 153 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−12.49594967402125641226130441183, −11.66395940209197304613461178617, −10.72529072213850473496056158414, −10.11100466934344249416021837933, −9.686913985103148389196636071308, −9.521753399418683578701524744309, −9.511288916198663122788067115574, −8.721897428937058715285939356035, −8.203038776544951088011641486497, −7.55386474433380282261805650620, −6.85808592357985285486251634332, −6.13843317655480644561020492224, −5.58593241357381875400456532428, −5.30755209249137639309945300775, −4.96418233993326260015093275755, −4.08039147382310237841234020896, −3.23726399256604874598239541314, −2.76599459262055859581424134014, −2.37358650132396160373017678788, −0.836379790650778760843858856357,
0.836379790650778760843858856357, 2.37358650132396160373017678788, 2.76599459262055859581424134014, 3.23726399256604874598239541314, 4.08039147382310237841234020896, 4.96418233993326260015093275755, 5.30755209249137639309945300775, 5.58593241357381875400456532428, 6.13843317655480644561020492224, 6.85808592357985285486251634332, 7.55386474433380282261805650620, 8.203038776544951088011641486497, 8.721897428937058715285939356035, 9.511288916198663122788067115574, 9.521753399418683578701524744309, 9.686913985103148389196636071308, 10.11100466934344249416021837933, 10.72529072213850473496056158414, 11.66395940209197304613461178617, 12.49594967402125641226130441183
