| L(s) = 1 | − 4·2-s − 21·5-s − 74·7-s + 64·8-s + 84·10-s − 270·11-s + 115·13-s + 296·14-s − 256·16-s − 1.72e3·17-s + 3.70e3·19-s + 1.08e3·22-s − 3.61e3·23-s + 3.12e3·25-s − 460·26-s − 1.12e3·29-s − 5.22e3·31-s + 6.88e3·34-s + 1.55e3·35-s + 1.98e4·37-s − 1.48e4·38-s − 1.34e3·40-s − 1.07e4·41-s + 1.97e4·43-s + 1.44e4·46-s − 9.98e3·47-s + 1.68e4·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.375·5-s − 0.570·7-s + 0.353·8-s + 0.265·10-s − 0.672·11-s + 0.188·13-s + 0.403·14-s − 1/4·16-s − 1.44·17-s + 2.35·19-s + 0.475·22-s − 1.42·23-s + 25-s − 0.133·26-s − 0.248·29-s − 0.977·31-s + 1.02·34-s + 0.214·35-s + 2.38·37-s − 1.66·38-s − 0.132·40-s − 0.999·41-s + 1.62·43-s + 1.00·46-s − 0.659·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9859821902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9859821902\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 21 T - 2684 T^{2} + 21 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 74 T - 11331 T^{2} + 74 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 270 T - 88151 T^{2} + 270 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 115 T - 358068 T^{2} - 115 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 861 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 1850 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3618 T + 6653581 T^{2} + 3618 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 1125 T - 19245524 T^{2} + 1125 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5228 T - 1297167 T^{2} + 5228 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 9917 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 10758 T - 121637 T^{2} + 10758 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 19714 T + 241633353 T^{2} - 19714 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9984 T - 129664751 T^{2} + 9984 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 36726 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26460 T - 14792699 T^{2} + 26460 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 53779 T + 2047584540 T^{2} - 53779 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12934 T - 1182836751 T^{2} - 12934 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4254 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 17521 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 36946 T - 1712049483 T^{2} - 36946 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 76416 T + 1900364413 T^{2} - 76416 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 45357 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 127574 T + 7687785219 T^{2} + 127574 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.95725833432204109789008224125, −11.78187572688829280480182021445, −11.10279581129843142820234485648, −10.67490370508215953986858267379, −10.12854709880607458746365962007, −9.599541798258637699933960877945, −9.192653552442823455242478236403, −8.740251602289111176055625549562, −7.963554817864361614039635942366, −7.70242345487695816721707267408, −7.02400997439735780104580356397, −6.59299272617491146376402800861, −5.50935043885459634915189713532, −5.43584676576139169529897608977, −4.23087236441569480597459129269, −3.90866927126287420103064819371, −2.85704337367496385546657868690, −2.31449377611395209955478322011, −1.10065351782441422392990979916, −0.43250884107805453715524685349,
0.43250884107805453715524685349, 1.10065351782441422392990979916, 2.31449377611395209955478322011, 2.85704337367496385546657868690, 3.90866927126287420103064819371, 4.23087236441569480597459129269, 5.43584676576139169529897608977, 5.50935043885459634915189713532, 6.59299272617491146376402800861, 7.02400997439735780104580356397, 7.70242345487695816721707267408, 7.963554817864361614039635942366, 8.740251602289111176055625549562, 9.192653552442823455242478236403, 9.599541798258637699933960877945, 10.12854709880607458746365962007, 10.67490370508215953986858267379, 11.10279581129843142820234485648, 11.78187572688829280480182021445, 11.95725833432204109789008224125
