| L(s) = 1 | − 2.66·2-s + 0.180·3-s + 5.10·4-s + 5-s − 0.482·6-s − 2.30·7-s − 8.27·8-s − 2.96·9-s − 2.66·10-s − 3.07·11-s + 0.923·12-s + 1.26·13-s + 6.13·14-s + 0.180·15-s + 11.8·16-s − 17-s + 7.90·18-s − 0.738·19-s + 5.10·20-s − 0.416·21-s + 8.19·22-s + 3.75·23-s − 1.49·24-s + 25-s − 3.37·26-s − 1.07·27-s − 11.7·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 0.104·3-s + 2.55·4-s + 0.447·5-s − 0.196·6-s − 0.869·7-s − 2.92·8-s − 0.989·9-s − 0.842·10-s − 0.927·11-s + 0.266·12-s + 0.351·13-s + 1.63·14-s + 0.0467·15-s + 2.95·16-s − 0.242·17-s + 1.86·18-s − 0.169·19-s + 1.14·20-s − 0.0908·21-s + 1.74·22-s + 0.782·23-s − 0.305·24-s + 0.200·25-s − 0.662·26-s − 0.207·27-s − 2.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 - 0.180T + 3T^{2} \) |
| 7 | \( 1 + 2.30T + 7T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 19 | \( 1 + 0.738T + 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 31 | \( 1 + 7.22T + 31T^{2} \) |
| 37 | \( 1 - 0.178T + 37T^{2} \) |
| 41 | \( 1 - 4.07T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 8.63T + 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 59 | \( 1 - 8.74T + 59T^{2} \) |
| 61 | \( 1 + 9.35T + 61T^{2} \) |
| 67 | \( 1 - 3.82T + 67T^{2} \) |
| 73 | \( 1 - 8.78T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 6.73T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.79742497869803547263446893261, −7.32013895093718969910664433887, −6.42037199231491300215308262010, −5.99371506020781226059972724385, −5.17963208496097273968969676807, −3.60307752316052712068156598932, −2.68871913900989925977398395295, −2.31733267289618934385484659223, −0.984654493260810941163690184472, 0,
0.984654493260810941163690184472, 2.31733267289618934385484659223, 2.68871913900989925977398395295, 3.60307752316052712068156598932, 5.17963208496097273968969676807, 5.99371506020781226059972724385, 6.42037199231491300215308262010, 7.32013895093718969910664433887, 7.79742497869803547263446893261
