| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 4·7-s + 8-s + 9-s + 10-s + 4·11-s + 12-s + 4·13-s + 4·14-s + 15-s + 16-s + 18-s + 4·19-s + 20-s + 4·21-s + 4·22-s − 8·23-s + 24-s + 25-s + 4·26-s + 27-s + 4·28-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 + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 1.10·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.872·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.755·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.05074807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.05074807\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.33174258017241, −12.11945526298797, −11.61969599991332, −11.19873289193339, −10.91246637550981, −10.21642666566173, −9.813915319930775, −9.277853665291824, −8.814578227241295, −8.288224698160298, −7.970225362066970, −7.536619082491514, −6.858456324936229, −6.428903891283008, −5.984191755473291, −5.383782425608411, −5.033955302498587, −4.396929171662967, −3.898051161496319, −3.613791485595202, −3.000799999961622, −2.135666453119328, −1.762374119044455, −1.448758798868972, −0.7469635362867544,
0.7469635362867544, 1.448758798868972, 1.762374119044455, 2.135666453119328, 3.000799999961622, 3.613791485595202, 3.898051161496319, 4.396929171662967, 5.033955302498587, 5.383782425608411, 5.984191755473291, 6.428903891283008, 6.858456324936229, 7.536619082491514, 7.970225362066970, 8.288224698160298, 8.814578227241295, 9.277853665291824, 9.813915319930775, 10.21642666566173, 10.91246637550981, 11.19873289193339, 11.61969599991332, 12.11945526298797, 12.33174258017241
