L(s) = 1 | − 7-s − 3·9-s + 4·11-s − 5·13-s − 3·17-s − 23-s + 8·29-s − 4·31-s + 37-s − 2·41-s − 10·43-s − 9·47-s + 49-s + 3·53-s + 8·59-s + 8·61-s + 3·63-s + 8·67-s + 4·71-s − 4·73-s − 4·77-s + 11·79-s + 9·81-s − 5·83-s + 5·89-s + 5·91-s − 5·97-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 9-s + 1.20·11-s − 1.38·13-s − 0.727·17-s − 0.208·23-s + 1.48·29-s − 0.718·31-s + 0.164·37-s − 0.312·41-s − 1.52·43-s − 1.31·47-s + 1/7·49-s + 0.412·53-s + 1.04·59-s + 1.02·61-s + 0.377·63-s + 0.977·67-s + 0.474·71-s − 0.468·73-s − 0.455·77-s + 1.23·79-s + 81-s − 0.548·83-s + 0.529·89-s + 0.524·91-s − 0.507·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.47393810875837, −14.09829477134421, −13.53537211601032, −12.95551649003872, −12.39906352712286, −11.90421413744587, −11.53880083805069, −11.12545959497013, −10.26579455157599, −9.906319609657752, −9.411062337257526, −8.798241802618904, −8.433569891268805, −7.836975428959306, −7.038492874983189, −6.604056385234769, −6.325797983613095, −5.380851055004019, −5.037078058201338, −4.340382362416881, −3.665711557068495, −3.089404018444118, −2.404181019605218, −1.848310955765167, −0.7851674731983728, 0,
0.7851674731983728, 1.848310955765167, 2.404181019605218, 3.089404018444118, 3.665711557068495, 4.340382362416881, 5.037078058201338, 5.380851055004019, 6.325797983613095, 6.604056385234769, 7.038492874983189, 7.836975428959306, 8.433569891268805, 8.798241802618904, 9.411062337257526, 9.906319609657752, 10.26579455157599, 11.12545959497013, 11.53880083805069, 11.90421413744587, 12.39906352712286, 12.95551649003872, 13.53537211601032, 14.09829477134421, 14.47393810875837