| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 2·13-s + 14-s − 15-s + 16-s + 6·17-s + 18-s + 19-s − 20-s + 21-s + 22-s + 24-s + 25-s + 2·26-s + 27-s + 28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.205830805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.205830805\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.46209894516634, −14.35992085997327, −13.65321439050832, −13.29547654428961, −12.44555712588268, −12.29477344955178, −11.65643944624380, −11.14583483209207, −10.55818426182842, −10.04380294191811, −9.395318437721913, −8.781487237168111, −8.214739051636566, −7.687162258589681, −7.314479658932340, −6.554426400978080, −5.970439005787710, −5.368692816738949, −4.694086148404337, −4.158260385600987, −3.462385340230230, −3.132136135908609, −2.294268583859572, −1.456892353633857, −0.8271498749453087,
0.8271498749453087, 1.456892353633857, 2.294268583859572, 3.132136135908609, 3.462385340230230, 4.158260385600987, 4.694086148404337, 5.368692816738949, 5.970439005787710, 6.554426400978080, 7.314479658932340, 7.687162258589681, 8.214739051636566, 8.781487237168111, 9.395318437721913, 10.04380294191811, 10.55818426182842, 11.14583483209207, 11.65643944624380, 12.29477344955178, 12.44555712588268, 13.29547654428961, 13.65321439050832, 14.35992085997327, 14.46209894516634
