| L(s) = 1 | − 2·3-s − 3·7-s + 9-s + 2·17-s − 2·19-s + 6·21-s − 9·23-s + 4·27-s + 2·29-s + 3·31-s + 12·37-s − 7·41-s + 8·43-s + 11·47-s + 2·49-s − 4·51-s + 8·53-s + 4·57-s + 10·59-s + 10·61-s − 3·63-s + 10·67-s + 18·69-s + 8·71-s + 11·73-s + 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.13·7-s + 1/3·9-s + 0.485·17-s − 0.458·19-s + 1.30·21-s − 1.87·23-s + 0.769·27-s + 0.371·29-s + 0.538·31-s + 1.97·37-s − 1.09·41-s + 1.21·43-s + 1.60·47-s + 2/7·49-s − 0.560·51-s + 1.09·53-s + 0.529·57-s + 1.30·59-s + 1.28·61-s − 0.377·63-s + 1.22·67-s + 2.16·69-s + 0.949·71-s + 1.28·73-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.006340654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.006340654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.42502515837489, −14.07747086045324, −13.46797430303774, −12.81856849020291, −12.46399810538305, −12.03075512124629, −11.47821308503752, −11.06282302944667, −10.20999711902998, −10.11244264942395, −9.559120021744289, −8.811816536066592, −8.188354611266616, −7.686139729374522, −6.757685365143430, −6.570948830650182, −5.876796208140296, −5.620344567401736, −4.895659581877423, −3.968292757257528, −3.837144075281119, −2.664983995767505, −2.312223072998185, −1.000789616685520, −0.4662505819161472,
0.4662505819161472, 1.000789616685520, 2.312223072998185, 2.664983995767505, 3.837144075281119, 3.968292757257528, 4.895659581877423, 5.620344567401736, 5.876796208140296, 6.570948830650182, 6.757685365143430, 7.686139729374522, 8.188354611266616, 8.811816536066592, 9.559120021744289, 10.11244264942395, 10.20999711902998, 11.06282302944667, 11.47821308503752, 12.03075512124629, 12.46399810538305, 12.81856849020291, 13.46797430303774, 14.07747086045324, 14.42502515837489
