| L(s) = 1 | − 2-s + 2·5-s + 4·7-s + 8-s − 2·10-s + 10·11-s − 4·14-s − 16-s + 4·17-s − 8·19-s − 10·22-s − 8·23-s − 7·25-s + 5·29-s − 3·31-s − 4·34-s + 8·35-s + 4·37-s + 8·38-s + 2·40-s − 2·43-s + 8·46-s + 6·47-s + 9·49-s + 7·50-s + 9·53-s + 20·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.894·5-s + 1.51·7-s + 0.353·8-s − 0.632·10-s + 3.01·11-s − 1.06·14-s − 1/4·16-s + 0.970·17-s − 1.83·19-s − 2.13·22-s − 1.66·23-s − 7/5·25-s + 0.928·29-s − 0.538·31-s − 0.685·34-s + 1.35·35-s + 0.657·37-s + 1.29·38-s + 0.316·40-s − 0.304·43-s + 1.17·46-s + 0.875·47-s + 9/7·49-s + 0.989·50-s + 1.23·53-s + 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.599010994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.599010994\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 11 T + 62 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 7 T - 34 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 7 T - 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−9.885091742322668318154652261159, −9.668074273613551459688038262444, −9.077556515309821629133765072707, −8.917908129767219229642457217633, −8.353306665935303634452004410265, −8.206947165046150323438989344873, −7.69186550839023917050464081808, −7.12022006626006840660572310824, −6.69597695111171852071264192433, −6.24613804831363455531798114053, −5.80513449795066747172927171629, −5.62675121393736725652141129291, −4.67272674170603994530873488218, −4.36105895330775565128051570096, −3.84287671461810534205376915146, −3.68862911514877503112798619708, −2.13974118550436682004688967592, −2.13762194134070572957538737961, −1.47261802301261065866000371488, −0.889780786249069642587307314762,
0.889780786249069642587307314762, 1.47261802301261065866000371488, 2.13762194134070572957538737961, 2.13974118550436682004688967592, 3.68862911514877503112798619708, 3.84287671461810534205376915146, 4.36105895330775565128051570096, 4.67272674170603994530873488218, 5.62675121393736725652141129291, 5.80513449795066747172927171629, 6.24613804831363455531798114053, 6.69597695111171852071264192433, 7.12022006626006840660572310824, 7.69186550839023917050464081808, 8.206947165046150323438989344873, 8.353306665935303634452004410265, 8.917908129767219229642457217633, 9.077556515309821629133765072707, 9.668074273613551459688038262444, 9.885091742322668318154652261159
