| L(s) = 1 | − 3-s − 3·7-s + 9-s − 5·11-s + 13-s + 5·17-s − 4·19-s + 3·21-s − 2·23-s − 27-s + 9·29-s + 3·31-s + 5·33-s − 10·37-s − 39-s − 12·41-s + 2·43-s − 9·47-s + 2·49-s − 5·51-s + 9·53-s + 4·57-s − 3·59-s + 7·61-s − 3·63-s + 9·67-s + 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 1.50·11-s + 0.277·13-s + 1.21·17-s − 0.917·19-s + 0.654·21-s − 0.417·23-s − 0.192·27-s + 1.67·29-s + 0.538·31-s + 0.870·33-s − 1.64·37-s − 0.160·39-s − 1.87·41-s + 0.304·43-s − 1.31·47-s + 2/7·49-s − 0.700·51-s + 1.23·53-s + 0.529·57-s − 0.390·59-s + 0.896·61-s − 0.377·63-s + 1.09·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.52934760257788, −13.81941450901686, −13.48216300765994, −12.95579457656505, −12.45093184112060, −12.14632849897432, −11.58429821344678, −10.78851234788335, −10.38298279127555, −9.989288810367870, −9.773624890365754, −8.708290944765454, −8.324968015050227, −7.888667091634409, −7.016028624554266, −6.663981429570822, −6.183930184310855, −5.399088519563102, −5.190423159867488, −4.424562940384925, −3.602154456386292, −3.156045719996091, −2.508155761132258, −1.690611171555460, −0.6919384125765738, 0,
0.6919384125765738, 1.690611171555460, 2.508155761132258, 3.156045719996091, 3.602154456386292, 4.424562940384925, 5.190423159867488, 5.399088519563102, 6.183930184310855, 6.663981429570822, 7.016028624554266, 7.888667091634409, 8.324968015050227, 8.708290944765454, 9.773624890365754, 9.989288810367870, 10.38298279127555, 10.78851234788335, 11.58429821344678, 12.14632849897432, 12.45093184112060, 12.95579457656505, 13.48216300765994, 13.81941450901686, 14.52934760257788
