| L(s) = 1 | + 2·2-s − 3·3-s + 3·4-s − 6·6-s + 2·7-s + 4·8-s + 6·9-s − 9·12-s + 4·14-s + 5·16-s + 12·18-s − 8·19-s − 6·21-s − 12·24-s − 2·25-s − 9·27-s + 6·28-s + 18·29-s + 6·32-s + 18·36-s − 16·38-s − 12·42-s + 4·43-s − 15·48-s − 11·49-s − 4·50-s − 18·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 3/2·4-s − 2.44·6-s + 0.755·7-s + 1.41·8-s + 2·9-s − 2.59·12-s + 1.06·14-s + 5/4·16-s + 2.82·18-s − 1.83·19-s − 1.30·21-s − 2.44·24-s − 2/5·25-s − 1.73·27-s + 1.13·28-s + 3.34·29-s + 1.06·32-s + 3·36-s − 2.59·38-s − 1.85·42-s + 0.609·43-s − 2.16·48-s − 1.57·49-s − 0.565·50-s − 2.47·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516113114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516113114\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−13.93865406482178534730604431979, −13.15489901751515059955542535017, −12.61561510748546313828220353693, −12.38098109882178152909384304628, −11.86405326257156272124322486984, −11.43843241121292595298890662752, −10.82738199385273632155144207714, −10.58246001556314574441209511667, −10.11430042684651296706286539675, −9.118119754959359622108277338484, −8.112262278083392832402569165284, −7.76232100624168519610049007081, −6.54564424096198188129094551060, −6.53910247390834776155477036816, −5.99522944989182225075727631750, −5.07148813490284239032473666909, −4.58512735759355047775599264373, −4.36832954540764876987581180628, −2.99063705687272663952860070260, −1.64369780833473190775040709017,
1.64369780833473190775040709017, 2.99063705687272663952860070260, 4.36832954540764876987581180628, 4.58512735759355047775599264373, 5.07148813490284239032473666909, 5.99522944989182225075727631750, 6.53910247390834776155477036816, 6.54564424096198188129094551060, 7.76232100624168519610049007081, 8.112262278083392832402569165284, 9.118119754959359622108277338484, 10.11430042684651296706286539675, 10.58246001556314574441209511667, 10.82738199385273632155144207714, 11.43843241121292595298890662752, 11.86405326257156272124322486984, 12.38098109882178152909384304628, 12.61561510748546313828220353693, 13.15489901751515059955542535017, 13.93865406482178534730604431979
