| L(s) = 1 | + 3-s − 4-s − 2·5-s + 2·7-s + 9-s − 12-s − 4·13-s − 2·15-s − 3·16-s + 2·17-s − 3·19-s + 2·20-s + 2·21-s − 4·23-s + 3·25-s + 4·27-s − 2·28-s − 3·29-s + 2·31-s − 4·35-s − 36-s + 4·37-s − 4·39-s − 3·41-s + 3·43-s − 2·45-s + 3·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 0.288·12-s − 1.10·13-s − 0.516·15-s − 3/4·16-s + 0.485·17-s − 0.688·19-s + 0.447·20-s + 0.436·21-s − 0.834·23-s + 3/5·25-s + 0.769·27-s − 0.377·28-s − 0.557·29-s + 0.359·31-s − 0.676·35-s − 1/6·36-s + 0.657·37-s − 0.640·39-s − 0.468·41-s + 0.457·43-s − 0.298·45-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2575 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2575 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6857761871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6857761871\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 103 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 14 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T + 40 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 9 T + 130 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 16 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−18.6313295751, −17.9962052447, −17.4896133915, −16.8730222353, −16.3820845843, −15.5731316882, −15.3282961344, −14.4821102124, −14.3834592578, −13.7263738526, −12.9342158948, −12.4472358856, −11.8442611700, −11.2774675003, −10.5707993844, −9.82834469887, −9.22939904256, −8.43153863965, −8.03104547970, −7.37519595607, −6.60287543086, −5.29298128274, −4.53054187920, −3.87224480694, −2.43726219077,
2.43726219077, 3.87224480694, 4.53054187920, 5.29298128274, 6.60287543086, 7.37519595607, 8.03104547970, 8.43153863965, 9.22939904256, 9.82834469887, 10.5707993844, 11.2774675003, 11.8442611700, 12.4472358856, 12.9342158948, 13.7263738526, 14.3834592578, 14.4821102124, 15.3282961344, 15.5731316882, 16.3820845843, 16.8730222353, 17.4896133915, 17.9962052447, 18.6313295751
