| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s − 4·11-s + 12-s − 2·13-s − 15-s + 16-s + 6·17-s + 18-s − 4·19-s − 20-s − 4·22-s + 24-s + 25-s − 2·26-s + 27-s − 2·29-s − 30-s + 32-s − 4·33-s + 6·34-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 − 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.852·22-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.371·29-s − 0.182·30-s + 0.176·32-s − 0.696·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.495210382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.495210382\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−15.70506621210930, −15.46806192333191, −14.69072255136574, −14.41663814610289, −13.89308565253634, −13.05100753761670, −12.62332074898766, −12.45332963366098, −11.47002922172573, −11.01797966778741, −10.33722192281353, −9.828581687317533, −9.180999770893100, −8.218907080700686, −7.882507637848893, −7.430180734568715, −6.671400986189318, −5.860669479401961, −5.267580314185694, −4.612253584949235, −3.941444104349471, −3.210555042013640, −2.627111190431267, −1.912937616846843, −0.6781922150984094,
0.6781922150984094, 1.912937616846843, 2.627111190431267, 3.210555042013640, 3.941444104349471, 4.612253584949235, 5.267580314185694, 5.860669479401961, 6.671400986189318, 7.430180734568715, 7.882507637848893, 8.218907080700686, 9.180999770893100, 9.828581687317533, 10.33722192281353, 11.01797966778741, 11.47002922172573, 12.45332963366098, 12.62332074898766, 13.05100753761670, 13.89308565253634, 14.41663814610289, 14.69072255136574, 15.46806192333191, 15.70506621210930
