| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s + 2·5-s − 4·6-s + 4·8-s + 3·9-s + 4·10-s − 10·11-s − 6·12-s + 2·13-s − 4·15-s + 5·16-s + 2·17-s + 6·18-s − 2·19-s + 6·20-s − 20·22-s − 10·23-s − 8·24-s − 7·25-s + 4·26-s − 4·27-s − 2·29-s − 8·30-s + 6·32-s + 20·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 1.41·8-s + 9-s + 1.26·10-s − 3.01·11-s − 1.73·12-s + 0.554·13-s − 1.03·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s − 0.458·19-s + 1.34·20-s − 4.26·22-s − 2.08·23-s − 1.63·24-s − 7/5·25-s + 0.784·26-s − 0.769·27-s − 0.371·29-s − 1.46·30-s + 1.06·32-s + 3.48·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14607684 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 10 T + 45 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 121 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 84 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 201 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 108 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 20 T + 224 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 144 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 138 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.920780859313957004596682522926, −7.81583843647811084181796443933, −7.46501685845457217385034664357, −7.32226953267307963064307580282, −6.41653785719251234976125057002, −6.22503762542794659896894840938, −5.88600844340545616742050182571, −5.78492248970426685553312608540, −5.29165725049836181685215368843, −5.10909179830418401912722999775, −4.60020722500704982587396409857, −4.30742093664611536269642371662, −3.63517565648508916993251297527, −3.39911339245620377301034545645, −2.59513001510396615719384651117, −2.44545681267662799709248243680, −1.67677616483415966989819735518, −1.61732694963993015620130702052, 0, 0,
1.61732694963993015620130702052, 1.67677616483415966989819735518, 2.44545681267662799709248243680, 2.59513001510396615719384651117, 3.39911339245620377301034545645, 3.63517565648508916993251297527, 4.30742093664611536269642371662, 4.60020722500704982587396409857, 5.10909179830418401912722999775, 5.29165725049836181685215368843, 5.78492248970426685553312608540, 5.88600844340545616742050182571, 6.22503762542794659896894840938, 6.41653785719251234976125057002, 7.32226953267307963064307580282, 7.46501685845457217385034664357, 7.81583843647811084181796443933, 7.920780859313957004596682522926
