| L(s) = 1 | − 2-s − 4-s − 2·7-s + 3·8-s + 4·11-s + 2·13-s + 2·14-s − 16-s + 4·17-s + 19-s − 4·22-s − 6·23-s − 2·26-s + 2·28-s + 6·29-s − 4·31-s − 5·32-s − 4·34-s + 10·37-s − 38-s + 10·41-s − 2·43-s − 4·44-s + 6·46-s − 6·47-s − 3·49-s − 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s + 1.20·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s + 0.970·17-s + 0.229·19-s − 0.852·22-s − 1.25·23-s − 0.392·26-s + 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.883·32-s − 0.685·34-s + 1.64·37-s − 0.162·38-s + 1.56·41-s − 0.304·43-s − 0.603·44-s + 0.884·46-s − 0.875·47-s − 3/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.107857073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107857073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.449478061028622315786843028520, −7.83667833810820080004463836281, −7.05055759660809826331982563360, −6.20016869246874032002970699005, −5.63368373308256164335975438652, −4.39866885171990376758924716324, −3.91292131323144132964594970635, −2.98182859075537613111881905749, −1.58572631934578227053492812984, −0.71640987977733752525952462098,
0.71640987977733752525952462098, 1.58572631934578227053492812984, 2.98182859075537613111881905749, 3.91292131323144132964594970635, 4.39866885171990376758924716324, 5.63368373308256164335975438652, 6.20016869246874032002970699005, 7.05055759660809826331982563360, 7.83667833810820080004463836281, 8.449478061028622315786843028520
