| L(s) = 1 | − 2·2-s − 4·3-s + 4-s − 5·5-s + 8·6-s − 3·7-s + 2·8-s + 8·9-s + 10·10-s − 5·11-s − 4·12-s − 3·13-s + 6·14-s + 20·15-s − 3·16-s − 16·18-s − 5·20-s + 12·21-s + 10·22-s − 5·23-s − 8·24-s + 16·25-s + 6·26-s − 12·27-s − 3·28-s − 40·30-s − 3·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s − 2.23·5-s + 3.26·6-s − 1.13·7-s + 0.707·8-s + 8/3·9-s + 3.16·10-s − 1.50·11-s − 1.15·12-s − 0.832·13-s + 1.60·14-s + 5.16·15-s − 3/4·16-s − 3.77·18-s − 1.11·20-s + 2.61·21-s + 2.13·22-s − 1.04·23-s − 1.63·24-s + 16/5·25-s + 1.17·26-s − 2.30·27-s − 0.566·28-s − 7.30·30-s − 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5170 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5170 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 8 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T - 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 15 T + 124 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 90 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 164 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T - 68 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−17.8469755854, −17.5900530504, −16.8661383015, −16.4505579301, −16.2461482738, −15.8818027294, −15.3391577074, −14.7873411580, −13.5858501088, −12.8430098630, −12.4484355418, −12.0222923483, −11.5323098707, −10.9300191086, −10.5881841884, −10.0543428415, −9.47428782586, −8.44527305830, −7.92601082051, −7.36764143454, −6.88614051004, −5.93805395449, −5.11380884851, −4.50832910358, −3.40824732257, 0, 0,
3.40824732257, 4.50832910358, 5.11380884851, 5.93805395449, 6.88614051004, 7.36764143454, 7.92601082051, 8.44527305830, 9.47428782586, 10.0543428415, 10.5881841884, 10.9300191086, 11.5323098707, 12.0222923483, 12.4484355418, 12.8430098630, 13.5858501088, 14.7873411580, 15.3391577074, 15.8818027294, 16.2461482738, 16.4505579301, 16.8661383015, 17.5900530504, 17.8469755854
