| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s − 7-s + 4·10-s − 11-s + 2·14-s − 4·16-s − 6·17-s − 19-s − 4·20-s + 2·22-s − 3·23-s − 25-s − 2·28-s − 6·29-s + 10·31-s + 8·32-s + 12·34-s + 2·35-s + 7·37-s + 2·38-s − 41-s + 4·43-s − 2·44-s + 6·46-s − 4·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s − 0.377·7-s + 1.26·10-s − 0.301·11-s + 0.534·14-s − 16-s − 1.45·17-s − 0.229·19-s − 0.894·20-s + 0.426·22-s − 0.625·23-s − 1/5·25-s − 0.377·28-s − 1.11·29-s + 1.79·31-s + 1.41·32-s + 2.05·34-s + 0.338·35-s + 1.15·37-s + 0.324·38-s − 0.156·41-s + 0.609·43-s − 0.301·44-s + 0.884·46-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3104494071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3104494071\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.39857394377969, −13.26850901413364, −12.50988494951440, −11.87294391962912, −11.58121434157226, −10.96105895739527, −10.74961470521289, −9.927687028991671, −9.804870745055361, −9.078062711731520, −8.673193056642158, −8.245307011610540, −7.654710208464009, −7.474904228473728, −6.658413179565574, −6.391030455634513, −5.664716674194930, −4.808788744808838, −4.238672610673109, −3.966720252642188, −2.997707203805252, −2.391854149515597, −1.865079359154462, −0.9186052440013064, −0.2691948874955428,
0.2691948874955428, 0.9186052440013064, 1.865079359154462, 2.391854149515597, 2.997707203805252, 3.966720252642188, 4.238672610673109, 4.808788744808838, 5.664716674194930, 6.391030455634513, 6.658413179565574, 7.474904228473728, 7.654710208464009, 8.245307011610540, 8.673193056642158, 9.078062711731520, 9.804870745055361, 9.927687028991671, 10.74961470521289, 10.96105895739527, 11.58121434157226, 11.87294391962912, 12.50988494951440, 13.26850901413364, 13.39857394377969
