| L(s) = 1 | − 2-s + 4-s − 6·7-s − 8-s + 9-s + 6·14-s + 16-s − 6·17-s − 18-s − 16·23-s + 7·25-s − 6·28-s + 3·31-s − 32-s + 6·34-s + 36-s − 9·41-s + 16·46-s + 5·47-s + 14·49-s − 7·50-s + 6·56-s − 3·62-s − 6·63-s + 64-s − 6·68-s − 2·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 2.26·7-s − 0.353·8-s + 1/3·9-s + 1.60·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 3.33·23-s + 7/5·25-s − 1.13·28-s + 0.538·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 1.40·41-s + 2.35·46-s + 0.729·47-s + 2·49-s − 0.989·50-s + 0.801·56-s − 0.381·62-s − 0.755·63-s + 1/8·64-s − 0.727·68-s − 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19328 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19328 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 151 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 13 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−10.38054930535642610294778834898, −9.976530211223208543334812798451, −9.735210243854786257274411550185, −9.058904278159911440766576690294, −8.572618757939796789896125442251, −7.981165275023874385498461849752, −7.13151998450410798976305136389, −6.64780760050782078422438221686, −6.30558806963369085080293627000, −5.73717596153813772377950917427, −4.51503217540298987344948064096, −3.79817144933635127086834888408, −3.01459010463696937239840734468, −2.10771662870371934397256343214, 0,
2.10771662870371934397256343214, 3.01459010463696937239840734468, 3.79817144933635127086834888408, 4.51503217540298987344948064096, 5.73717596153813772377950917427, 6.30558806963369085080293627000, 6.64780760050782078422438221686, 7.13151998450410798976305136389, 7.981165275023874385498461849752, 8.572618757939796789896125442251, 9.058904278159911440766576690294, 9.735210243854786257274411550185, 9.976530211223208543334812798451, 10.38054930535642610294778834898
