| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 9-s − 4·11-s − 2·12-s − 4·16-s − 6·17-s + 2·18-s − 4·19-s − 8·22-s − 25-s − 27-s − 8·32-s + 4·33-s − 12·34-s + 2·36-s − 8·38-s + 2·41-s − 3·43-s − 8·44-s + 4·48-s − 5·49-s − 2·50-s + 6·51-s − 2·54-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 16-s − 1.45·17-s + 0.471·18-s − 0.917·19-s − 1.70·22-s − 1/5·25-s − 0.192·27-s − 1.41·32-s + 0.696·33-s − 2.05·34-s + 1/3·36-s − 1.29·38-s + 0.312·41-s − 0.457·43-s − 1.20·44-s + 0.577·48-s − 5/7·49-s − 0.282·50-s + 0.840·51-s − 0.272·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.346597844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.346597844\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−7.55924990620496184137476096961, −7.25957583810710511443942132683, −6.70480175545812344884697160110, −6.38146092617065090512491111802, −5.98267016928500392756278367154, −5.62418874255260191555146770277, −5.09479728845962694945303688014, −4.67774424948497052360074556201, −4.40811726489878898134422739260, −3.99677988477541234273761249388, −3.20394418964201884302398371342, −2.90246733082985853840017428502, −2.16845695708643779804492084863, −1.80045417741345461959333256537, −0.36154590492937417568250507979,
0.36154590492937417568250507979, 1.80045417741345461959333256537, 2.16845695708643779804492084863, 2.90246733082985853840017428502, 3.20394418964201884302398371342, 3.99677988477541234273761249388, 4.40811726489878898134422739260, 4.67774424948497052360074556201, 5.09479728845962694945303688014, 5.62418874255260191555146770277, 5.98267016928500392756278367154, 6.38146092617065090512491111802, 6.70480175545812344884697160110, 7.25957583810710511443942132683, 7.55924990620496184137476096961
