| L(s) = 1 | − 2·2-s − 2·3-s + 4-s − 5·5-s + 4·6-s − 7·7-s + 2·8-s + 4·9-s + 10·10-s − 3·11-s − 2·12-s − 6·13-s + 14·14-s + 10·15-s − 3·16-s + 2·17-s − 8·18-s − 19-s − 5·20-s + 14·21-s + 6·22-s + 23-s − 4·24-s + 12·25-s + 12·26-s − 5·27-s − 7·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s − 2.23·5-s + 1.63·6-s − 2.64·7-s + 0.707·8-s + 4/3·9-s + 3.16·10-s − 0.904·11-s − 0.577·12-s − 1.66·13-s + 3.74·14-s + 2.58·15-s − 3/4·16-s + 0.485·17-s − 1.88·18-s − 0.229·19-s − 1.11·20-s + 3.05·21-s + 1.27·22-s + 0.208·23-s − 0.816·24-s + 12/5·25-s + 2.35·26-s − 0.962·27-s − 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9426 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9426 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T + p T^{2} ) \) |
| 1571 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 8 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - T + 43 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14 T + 104 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 34 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 44 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 17 T + 176 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T - 73 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 110 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 91 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 43 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 118 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 70 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−16.9280409100, −16.7270579676, −16.2383652134, −15.9386926118, −15.5340722079, −15.2190405433, −14.3153257298, −13.3460580047, −12.7598642633, −12.4417665843, −12.2274255379, −11.5631537173, −10.7315453943, −10.3457161827, −10.0450271194, −9.33545771317, −8.95201571471, −7.88280081405, −7.56633606443, −7.07102633146, −6.60702909629, −5.49863254975, −4.69780522959, −3.80521693408, −3.11258961973, 0, 0,
3.11258961973, 3.80521693408, 4.69780522959, 5.49863254975, 6.60702909629, 7.07102633146, 7.56633606443, 7.88280081405, 8.95201571471, 9.33545771317, 10.0450271194, 10.3457161827, 10.7315453943, 11.5631537173, 12.2274255379, 12.4417665843, 12.7598642633, 13.3460580047, 14.3153257298, 15.2190405433, 15.5340722079, 15.9386926118, 16.2383652134, 16.7270579676, 16.9280409100
