| L(s) = 1 | + 3·2-s + 4·4-s + 6·5-s − 5·7-s + 3·8-s + 18·10-s − 3·13-s − 15·14-s + 3·16-s − 3·17-s − 9·19-s + 24·20-s + 17·25-s − 9·26-s − 20·28-s + 9·29-s − 6·31-s + 6·32-s − 9·34-s − 30·35-s − 7·37-s − 27·38-s + 18·40-s − 3·41-s − 43-s + 18·49-s + 51·50-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s + 2.68·5-s − 1.88·7-s + 1.06·8-s + 5.69·10-s − 0.832·13-s − 4.00·14-s + 3/4·16-s − 0.727·17-s − 2.06·19-s + 5.36·20-s + 17/5·25-s − 1.76·26-s − 3.77·28-s + 1.67·29-s − 1.07·31-s + 1.06·32-s − 1.54·34-s − 5.07·35-s − 1.15·37-s − 4.37·38-s + 2.84·40-s − 0.468·41-s − 0.152·43-s + 18/7·49-s + 7.21·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.161687034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.161687034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15 T + 128 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−12.80389300771747457324342231409, −12.76162064228188467154084977381, −12.38251671581157514297850003327, −11.67531767153783021823022308673, −10.61745912684713716323891137596, −10.37798055714551652126354418374, −9.873579597725457193422854875596, −9.703844744779360319034302167829, −8.715092273493901535443830730038, −8.708631301806503453496507094623, −6.92615712046326489284773316905, −6.89213041171976789131685994296, −6.20616212865813750100494825717, −5.89657219822988426669678908810, −5.42297074508985339728239871501, −4.85130786653000608853382043751, −4.09932339155468257746074422346, −3.39720133514273382413083312497, −2.41184879641827223592830948461, −2.21997194182799098016930422096,
2.21997194182799098016930422096, 2.41184879641827223592830948461, 3.39720133514273382413083312497, 4.09932339155468257746074422346, 4.85130786653000608853382043751, 5.42297074508985339728239871501, 5.89657219822988426669678908810, 6.20616212865813750100494825717, 6.89213041171976789131685994296, 6.92615712046326489284773316905, 8.708631301806503453496507094623, 8.715092273493901535443830730038, 9.703844744779360319034302167829, 9.873579597725457193422854875596, 10.37798055714551652126354418374, 10.61745912684713716323891137596, 11.67531767153783021823022308673, 12.38251671581157514297850003327, 12.76162064228188467154084977381, 12.80389300771747457324342231409
