| L(s) = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s − 4·7-s + 8-s + 9-s + 10-s + 4·11-s + 2·12-s − 2·13-s − 4·14-s + 2·15-s + 16-s − 2·17-s + 18-s + 4·19-s + 20-s − 8·21-s + 4·22-s − 4·23-s + 2·24-s + 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 + 0.447·5-s + 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.577·12-s − 0.554·13-s − 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 1.74·21-s + 0.852·22-s − 0.834·23-s + 0.408·24-s + 1/5·25-s − 0.392·26-s − 0.769·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.90809076862174, −13.73156988680064, −13.06421308081998, −12.71351352448939, −12.20380001808805, −11.68411205320039, −11.24824214262590, −10.25930751922463, −10.05458663221576, −9.443427512213674, −9.173557346685241, −8.646760220669131, −8.009490421005505, −7.282291514794970, −6.958575842904407, −6.377821806724283, −5.893042152994868, −5.457307304189248, −4.316242218204405, −4.258046149197597, −3.278089361696878, −3.144154452746842, −2.525328372926191, −1.898081828591398, −1.114916025005716, 0,
1.114916025005716, 1.898081828591398, 2.525328372926191, 3.144154452746842, 3.278089361696878, 4.258046149197597, 4.316242218204405, 5.457307304189248, 5.893042152994868, 6.377821806724283, 6.958575842904407, 7.282291514794970, 8.009490421005505, 8.646760220669131, 9.173557346685241, 9.443427512213674, 10.05458663221576, 10.25930751922463, 11.24824214262590, 11.68411205320039, 12.20380001808805, 12.71351352448939, 13.06421308081998, 13.73156988680064, 13.90809076862174
