L(s) = 1 | − 2-s + 2·3-s − 4-s + 3·5-s − 2·6-s + 4·7-s + 3·8-s + 9-s − 3·10-s − 4·11-s − 2·12-s − 2·13-s − 4·14-s + 6·15-s − 16-s − 3·17-s − 18-s + 19-s − 3·20-s + 8·21-s + 4·22-s + 7·23-s + 6·24-s + 4·25-s + 2·26-s − 4·27-s − 4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s + 1.34·5-s − 0.816·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.948·10-s − 1.20·11-s − 0.577·12-s − 0.554·13-s − 1.06·14-s + 1.54·15-s − 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.229·19-s − 0.670·20-s + 1.74·21-s + 0.852·22-s + 1.45·23-s + 1.22·24-s + 4/5·25-s + 0.392·26-s − 0.769·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31939 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31939 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 - T \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.01059877269697, −14.69842576144412, −14.31797211854825, −13.70175671018537, −13.35622443062187, −13.07856829739937, −12.27261257841658, −11.33465651126155, −10.81335008084364, −10.43445922657055, −9.761708767706927, −9.307602977712748, −8.848880599033317, −8.333782731288707, −8.000076605589945, −7.316415523592087, −6.846879764836664, −5.519063264026636, −5.360023189903999, −4.726282611499061, −4.101916375998842, −2.915192671732627, −2.539758292997555, −1.740751495393528, −1.358821273105602, 0,
1.358821273105602, 1.740751495393528, 2.539758292997555, 2.915192671732627, 4.101916375998842, 4.726282611499061, 5.360023189903999, 5.519063264026636, 6.846879764836664, 7.316415523592087, 8.000076605589945, 8.333782731288707, 8.848880599033317, 9.307602977712748, 9.761708767706927, 10.43445922657055, 10.81335008084364, 11.33465651126155, 12.27261257841658, 13.07856829739937, 13.35622443062187, 13.70175671018537, 14.31797211854825, 14.69842576144412, 15.01059877269697