L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s + 7-s + 3·8-s − 2·9-s − 10-s − 5·11-s + 12-s − 5·13-s − 14-s − 15-s − 16-s − 4·17-s + 2·18-s − 4·19-s − 20-s − 21-s + 5·22-s + 6·23-s − 3·24-s − 4·25-s + 5·26-s + 5·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s − 2/3·9-s − 0.316·10-s − 1.50·11-s + 0.288·12-s − 1.38·13-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.970·17-s + 0.471·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s + 1.06·22-s + 1.25·23-s − 0.612·24-s − 4/5·25-s + 0.980·26-s + 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−11.80461118678257977742116101174, −10.64028007338165891977785822018, −10.15824528112640096033599437670, −8.906691467674992674253555806226, −8.125538957772544309632418278080, −6.93906844697635318855785134503, −5.37765856689412319458680890340, −4.73798046978744347074774135421, −2.41583763287982840154273478188, 0,
2.41583763287982840154273478188, 4.73798046978744347074774135421, 5.37765856689412319458680890340, 6.93906844697635318855785134503, 8.125538957772544309632418278080, 8.906691467674992674253555806226, 10.15824528112640096033599437670, 10.64028007338165891977785822018, 11.80461118678257977742116101174