| L(s) = 1 | + 2-s − 3·3-s − 3·6-s − 5·7-s − 8-s + 3·9-s + 4·13-s − 5·14-s − 16-s − 2·17-s + 3·18-s + 2·19-s + 15·21-s − 23-s + 3·24-s + 4·26-s − 2·29-s − 10·31-s − 2·34-s − 8·37-s + 2·38-s − 12·39-s − 6·41-s + 15·42-s − 10·43-s − 46-s + 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s − 1.22·6-s − 1.88·7-s − 0.353·8-s + 9-s + 1.10·13-s − 1.33·14-s − 1/4·16-s − 0.485·17-s + 0.707·18-s + 0.458·19-s + 3.27·21-s − 0.208·23-s + 0.612·24-s + 0.784·26-s − 0.371·29-s − 1.79·31-s − 0.342·34-s − 1.31·37-s + 0.324·38-s − 1.92·39-s − 0.937·41-s + 2.31·42-s − 1.52·43-s − 0.147·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4459667053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4459667053\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−12.07437486845391358146796619697, −11.26948205131981146021901079043, −10.98607620875950455439954139780, −10.29299744666972837081543763376, −10.26461302921956060053819418277, −9.326217105439490579737873897000, −9.164264949147969639751508058194, −8.609091153860219118402845834657, −7.82328455295202986781267255488, −6.95582827991260943650974203612, −6.81380190649926851391580438902, −6.16264184187355367482836555314, −5.96941015876984693134168855238, −5.34156699539499772301423828215, −5.10462796917415574802782240317, −4.13356913470701269888723617041, −3.45868192196357096712005263972, −3.31740197068921615268708359396, −1.97703365690165919918700415792, −0.44724677108090400808069923914,
0.44724677108090400808069923914, 1.97703365690165919918700415792, 3.31740197068921615268708359396, 3.45868192196357096712005263972, 4.13356913470701269888723617041, 5.10462796917415574802782240317, 5.34156699539499772301423828215, 5.96941015876984693134168855238, 6.16264184187355367482836555314, 6.81380190649926851391580438902, 6.95582827991260943650974203612, 7.82328455295202986781267255488, 8.609091153860219118402845834657, 9.164264949147969639751508058194, 9.326217105439490579737873897000, 10.26461302921956060053819418277, 10.29299744666972837081543763376, 10.98607620875950455439954139780, 11.26948205131981146021901079043, 12.07437486845391358146796619697
