| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 4·5-s + 4·6-s − 4·8-s + 8·10-s + 11-s − 4·12-s − 13-s + 8·15-s + 8·16-s − 17-s − 8·20-s − 2·22-s − 3·23-s + 8·24-s + 6·25-s + 2·26-s + 2·27-s + 3·29-s − 16·30-s − 6·31-s − 8·32-s − 2·33-s + 2·34-s + 37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s − 1.41·8-s + 2.52·10-s + 0.301·11-s − 1.15·12-s − 0.277·13-s + 2.06·15-s + 2·16-s − 0.242·17-s − 1.78·20-s − 0.426·22-s − 0.625·23-s + 1.63·24-s + 6/5·25-s + 0.392·26-s + 0.384·27-s + 0.557·29-s − 2.92·30-s − 1.07·31-s − 1.41·32-s − 0.348·33-s + 0.342·34-s + 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1637 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1637 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 1637 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 44 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 43 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T - 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 41 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 103 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 15 T + 182 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 88 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−19.4127301786, −18.8652734850, −18.1575153949, −17.9504441633, −17.3170342178, −16.8882757907, −16.2287255539, −15.9442576661, −15.2600498235, −14.7924726960, −14.0900448940, −12.7630822331, −12.2774990606, −11.8274600643, −11.2209241533, −11.0486382202, −10.0395941485, −9.33596011448, −8.66686807455, −7.94841399901, −7.50797586611, −6.45240303254, −5.80324006253, −4.53626004875, −3.32634422552, 0,
3.32634422552, 4.53626004875, 5.80324006253, 6.45240303254, 7.50797586611, 7.94841399901, 8.66686807455, 9.33596011448, 10.0395941485, 11.0486382202, 11.2209241533, 11.8274600643, 12.2774990606, 12.7630822331, 14.0900448940, 14.7924726960, 15.2600498235, 15.9442576661, 16.2287255539, 16.8882757907, 17.3170342178, 17.9504441633, 18.1575153949, 18.8652734850, 19.4127301786
