| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 2·11-s + 12-s − 7·13-s + 15-s + 16-s − 3·17-s + 18-s + 6·19-s + 20-s − 2·22-s + 3·23-s + 24-s + 25-s − 7·26-s + 27-s + 29-s + 30-s − 3·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 1.94·13-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s − 0.426·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s − 1.37·26-s + 0.192·27-s + 0.185·29-s + 0.182·30-s − 0.538·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.544177183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.544177183\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.70118294290543, −14.13743038949439, −13.75733122275024, −13.21186228663772, −12.71186495592533, −12.27147329587105, −11.75760666632693, −11.08871821880840, −10.46241435231163, −10.04151261300499, −9.370209397733832, −9.103611102738387, −8.230170780476072, −7.577259448647346, −7.172094388715973, −6.823015022341544, −5.802763193921996, −5.363847663782851, −4.840145689602435, −4.310882421204973, −3.457712179783261, −2.750385240728468, −2.447203968601046, −1.712906174990578, −0.6295535842134685,
0.6295535842134685, 1.712906174990578, 2.447203968601046, 2.750385240728468, 3.457712179783261, 4.310882421204973, 4.840145689602435, 5.363847663782851, 5.802763193921996, 6.823015022341544, 7.172094388715973, 7.577259448647346, 8.230170780476072, 9.103611102738387, 9.370209397733832, 10.04151261300499, 10.46241435231163, 11.08871821880840, 11.75760666632693, 12.27147329587105, 12.71186495592533, 13.21186228663772, 13.75733122275024, 14.13743038949439, 14.70118294290543
