| L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 3·8-s + 9-s + 10-s + 3·11-s − 3·13-s − 2·14-s + 16-s − 17-s − 18-s − 6·19-s − 20-s − 3·22-s − 23-s − 2·25-s + 3·26-s + 2·28-s − 2·29-s + 3·31-s + 32-s + 34-s − 2·35-s + 36-s − 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s − 0.832·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.639·22-s − 0.208·23-s − 2/5·25-s + 0.588·26-s + 0.377·28-s − 0.371·29-s + 0.538·31-s + 0.176·32-s + 0.171·34-s − 0.338·35-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1935 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1935 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4690946715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4690946715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 30 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - T - 36 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 160 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−18.9641005199, −18.3517461462, −17.7835924166, −17.3514188172, −16.9529405744, −16.3014711788, −15.5409151911, −15.2077926146, −14.6509093727, −14.1836599372, −13.2901403907, −12.5217288047, −11.9886205160, −11.5258180006, −10.9406130681, −10.1696962899, −9.48410016152, −8.83072359807, −8.28755135037, −7.49691530016, −6.75777936318, −6.05289371361, −4.78833523780, −3.89775393846, −2.25840737714,
2.25840737714, 3.89775393846, 4.78833523780, 6.05289371361, 6.75777936318, 7.49691530016, 8.28755135037, 8.83072359807, 9.48410016152, 10.1696962899, 10.9406130681, 11.5258180006, 11.9886205160, 12.5217288047, 13.2901403907, 14.1836599372, 14.6509093727, 15.2077926146, 15.5409151911, 16.3014711788, 16.9529405744, 17.3514188172, 17.7835924166, 18.3517461462, 18.9641005199
