| L(s) = 1 | − 2-s + 2·3-s − 2·6-s + 4·7-s + 8-s + 3·9-s + 2·11-s + 13-s − 4·14-s − 16-s + 6·17-s − 3·18-s − 7·19-s + 8·21-s − 2·22-s + 2·24-s + 5·25-s − 26-s + 10·27-s − 9·29-s + 16·31-s + 4·33-s − 6·34-s − 20·37-s + 7·38-s + 2·39-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 0.816·6-s + 1.51·7-s + 0.353·8-s + 9-s + 0.603·11-s + 0.277·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s − 0.707·18-s − 1.60·19-s + 1.74·21-s − 0.426·22-s + 0.408·24-s + 25-s − 0.196·26-s + 1.92·27-s − 1.67·29-s + 2.87·31-s + 0.696·33-s − 1.02·34-s − 3.28·37-s + 1.13·38-s + 0.320·39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.243616550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.243616550\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−11.22682234217004666619911428810, −10.88001311492037414070644940726, −10.40060935443841962456662611276, −10.04888732771626787755320277732, −9.600714985020813527948544248553, −8.976529768976431548031177304582, −8.467779840250957616275657126665, −8.285603398092119200046154165400, −8.234048812911613774936302853707, −7.40561654795960395919066679000, −6.84542093230910216464929138403, −6.62770548705313872563619376033, −5.56151146100428010617863353430, −5.13831236350976290999180483901, −4.40904864180515439500091636305, −4.08903152934446920283427507627, −3.27876004991778180731223911297, −2.59351915792078184431253759349, −1.67531807233687475215544391971, −1.24521425287284537056541760796,
1.24521425287284537056541760796, 1.67531807233687475215544391971, 2.59351915792078184431253759349, 3.27876004991778180731223911297, 4.08903152934446920283427507627, 4.40904864180515439500091636305, 5.13831236350976290999180483901, 5.56151146100428010617863353430, 6.62770548705313872563619376033, 6.84542093230910216464929138403, 7.40561654795960395919066679000, 8.234048812911613774936302853707, 8.285603398092119200046154165400, 8.467779840250957616275657126665, 8.976529768976431548031177304582, 9.600714985020813527948544248553, 10.04888732771626787755320277732, 10.40060935443841962456662611276, 10.88001311492037414070644940726, 11.22682234217004666619911428810
