| L(s) = 1 | − 11-s − 2·13-s + 6·17-s + 4·19-s − 4·23-s − 6·29-s + 8·31-s + 2·37-s − 2·41-s + 4·43-s + 12·47-s − 7·49-s − 2·53-s + 4·59-s − 10·61-s − 16·67-s + 8·71-s − 14·73-s − 8·79-s + 4·83-s − 10·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.301·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.834·23-s − 1.11·29-s + 1.43·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 1.75·47-s − 49-s − 0.274·53-s + 0.520·59-s − 1.28·61-s − 1.95·67-s + 0.949·71-s − 1.63·73-s − 0.900·79-s + 0.439·83-s − 1.05·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.95012037863683, −14.58860052300546, −13.94824040957785, −13.63469376006681, −12.98192172785398, −12.27758706965098, −12.03494227146982, −11.54435738525246, −10.76357960250734, −10.29590448366618, −9.728605408736568, −9.421666836585223, −8.646302575385272, −7.921973867470317, −7.633487779313294, −7.125937341792001, −6.273767268888799, −5.675294016750433, −5.338855325278491, −4.490651113400702, −3.977616239269208, −3.071811715635522, −2.744157344655301, −1.727750737325850, −1.031628594892046, 0,
1.031628594892046, 1.727750737325850, 2.744157344655301, 3.071811715635522, 3.977616239269208, 4.490651113400702, 5.338855325278491, 5.675294016750433, 6.273767268888799, 7.125937341792001, 7.633487779313294, 7.921973867470317, 8.646302575385272, 9.421666836585223, 9.728605408736568, 10.29590448366618, 10.76357960250734, 11.54435738525246, 12.03494227146982, 12.27758706965098, 12.98192172785398, 13.63469376006681, 13.94824040957785, 14.58860052300546, 14.95012037863683
