| L(s) = 1 | − 2·2-s + 2·4-s + 5-s + 7-s − 2·10-s + 2·13-s − 2·14-s − 4·16-s + 17-s + 7·19-s + 2·20-s − 5·23-s + 25-s − 4·26-s + 2·28-s + 3·29-s − 4·31-s + 8·32-s − 2·34-s + 35-s − 2·37-s − 14·38-s − 12·41-s + 43-s + 10·46-s + 4·47-s + 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s + 0.377·7-s − 0.632·10-s + 0.554·13-s − 0.534·14-s − 16-s + 0.242·17-s + 1.60·19-s + 0.447·20-s − 1.04·23-s + 1/5·25-s − 0.784·26-s + 0.377·28-s + 0.557·29-s − 0.718·31-s + 1.41·32-s − 0.342·34-s + 0.169·35-s − 0.328·37-s − 2.27·38-s − 1.87·41-s + 0.152·43-s + 1.47·46-s + 0.583·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−15.44892668792593, −14.46028366068095, −14.08972208255630, −13.58902032061106, −13.17583758352650, −12.18091962789594, −11.91040411970756, −11.22968400725957, −10.76293835000682, −10.18072720356840, −9.763844094944723, −9.341491034217390, −8.662773203816934, −8.247192123572037, −7.775140394880535, −7.072018638742970, −6.725811857723793, −5.787339471657950, −5.381778047866521, −4.627512822590192, −3.826028249386005, −3.097959296201521, −2.233613967360924, −1.535583688650142, −1.046658679007154, 0,
1.046658679007154, 1.535583688650142, 2.233613967360924, 3.097959296201521, 3.826028249386005, 4.627512822590192, 5.381778047866521, 5.787339471657950, 6.725811857723793, 7.072018638742970, 7.775140394880535, 8.247192123572037, 8.662773203816934, 9.341491034217390, 9.763844094944723, 10.18072720356840, 10.76293835000682, 11.22968400725957, 11.91040411970756, 12.18091962789594, 13.17583758352650, 13.58902032061106, 14.08972208255630, 14.46028366068095, 15.44892668792593
