| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 4·5-s + 9·6-s − 3·7-s − 3·8-s + 2·9-s + 12·10-s − 4·11-s − 12·12-s − 4·13-s + 9·14-s + 12·15-s + 3·16-s − 5·17-s − 6·18-s − 4·19-s − 16·20-s + 9·21-s + 12·22-s − 12·23-s + 9·24-s + 6·25-s + 12·26-s + 6·27-s − 12·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 1.78·5-s + 3.67·6-s − 1.13·7-s − 1.06·8-s + 2/3·9-s + 3.79·10-s − 1.20·11-s − 3.46·12-s − 1.10·13-s + 2.40·14-s + 3.09·15-s + 3/4·16-s − 1.21·17-s − 1.41·18-s − 0.917·19-s − 3.57·20-s + 1.96·21-s + 2.55·22-s − 2.50·23-s + 1.83·24-s + 6/5·25-s + 2.35·26-s + 1.15·27-s − 2.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105353 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105353 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 137 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| 769 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 14 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 23 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 61 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T - 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 55 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T - 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9 T + 140 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 93 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T - 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T + 129 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−14.8939519457, −14.5126702642, −13.5770119791, −13.1655430373, −12.4193248577, −12.2798145364, −11.9514965787, −11.3383476229, −11.1351664999, −10.5680226662, −10.1557551590, −10.0168151477, −9.23745880475, −8.80191825321, −8.31091207267, −7.84259898484, −7.53917018534, −6.99147143928, −6.35538801266, −5.91085694349, −5.33820602490, −4.51722084923, −3.99684181833, −3.08147959304, −2.10343773968, 0, 0, 0,
2.10343773968, 3.08147959304, 3.99684181833, 4.51722084923, 5.33820602490, 5.91085694349, 6.35538801266, 6.99147143928, 7.53917018534, 7.84259898484, 8.31091207267, 8.80191825321, 9.23745880475, 10.0168151477, 10.1557551590, 10.5680226662, 11.1351664999, 11.3383476229, 11.9514965787, 12.2798145364, 12.4193248577, 13.1655430373, 13.5770119791, 14.5126702642, 14.8939519457
