| L(s) = 1 | + 2-s + 2·4-s − 5-s + 5·8-s − 10-s + 5·11-s + 5·13-s + 5·16-s − 6·17-s + 2·19-s − 2·20-s + 5·22-s + 3·23-s + 5·25-s + 5·26-s − 29-s + 10·32-s − 6·34-s + 6·37-s + 2·38-s − 5·40-s − 5·41-s + 43-s + 10·44-s + 3·46-s + 5·50-s + 10·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s − 0.447·5-s + 1.76·8-s − 0.316·10-s + 1.50·11-s + 1.38·13-s + 5/4·16-s − 1.45·17-s + 0.458·19-s − 0.447·20-s + 1.06·22-s + 0.625·23-s + 25-s + 0.980·26-s − 0.185·29-s + 1.76·32-s − 1.02·34-s + 0.986·37-s + 0.324·38-s − 0.790·40-s − 0.780·41-s + 0.152·43-s + 1.50·44-s + 0.442·46-s + 0.707·50-s + 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.762179656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.762179656\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 9 T - 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−9.827786587342908299610877344975, −9.425299576344295973716770389234, −8.857317096816128570703841642266, −8.716782766916085427578498064222, −8.123691234363019259183943381850, −7.83496120326585387631539143519, −7.07778076857527006054714210820, −6.94763899406371646143018562965, −6.53391051084245554000618588398, −6.36265458883412573163699775731, −5.61560984657135122140564774575, −5.12198921335811558057689442127, −4.74613736808218236622065223945, −4.04459688183498998307446129871, −3.85972035716444058078916352829, −3.59274358122106589588075045065, −2.53259380379067120516362596354, −2.34070684351117564245518225379, −1.23422931124403506956313915831, −1.13297985530683275244795338550,
1.13297985530683275244795338550, 1.23422931124403506956313915831, 2.34070684351117564245518225379, 2.53259380379067120516362596354, 3.59274358122106589588075045065, 3.85972035716444058078916352829, 4.04459688183498998307446129871, 4.74613736808218236622065223945, 5.12198921335811558057689442127, 5.61560984657135122140564774575, 6.36265458883412573163699775731, 6.53391051084245554000618588398, 6.94763899406371646143018562965, 7.07778076857527006054714210820, 7.83496120326585387631539143519, 8.123691234363019259183943381850, 8.716782766916085427578498064222, 8.857317096816128570703841642266, 9.425299576344295973716770389234, 9.827786587342908299610877344975
