| L(s) = 1 | − 0.600·5-s + 7-s − 1.60·11-s + 0.330·13-s + 1.44·17-s − 2.57·19-s + 1.84·23-s − 4.63·25-s − 3.51·29-s + 9.62·31-s − 0.600·35-s + 0.600·37-s − 6.62·41-s − 3.62·43-s + 3.90·47-s + 49-s + 9.27·53-s + 0.961·55-s − 13.8·59-s − 5.18·61-s − 0.198·65-s + 11.8·67-s − 4.17·71-s + 4.13·73-s − 1.60·77-s − 8.13·79-s − 5.57·83-s + ⋯ |
| L(s) = 1 | − 0.268·5-s + 0.377·7-s − 0.482·11-s + 0.0916·13-s + 0.351·17-s − 0.591·19-s + 0.385·23-s − 0.927·25-s − 0.653·29-s + 1.72·31-s − 0.101·35-s + 0.0987·37-s − 1.03·41-s − 0.553·43-s + 0.570·47-s + 0.142·49-s + 1.27·53-s + 0.129·55-s − 1.80·59-s − 0.664·61-s − 0.0246·65-s + 1.44·67-s − 0.496·71-s + 0.484·73-s − 0.182·77-s − 0.914·79-s − 0.612·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 0.600T + 5T^{2} \) |
| 11 | \( 1 + 1.60T + 11T^{2} \) |
| 13 | \( 1 - 0.330T + 13T^{2} \) |
| 17 | \( 1 - 1.44T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 - 1.84T + 23T^{2} \) |
| 29 | \( 1 + 3.51T + 29T^{2} \) |
| 31 | \( 1 - 9.62T + 31T^{2} \) |
| 37 | \( 1 - 0.600T + 37T^{2} \) |
| 41 | \( 1 + 6.62T + 41T^{2} \) |
| 43 | \( 1 + 3.62T + 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 - 9.27T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 - 4.13T + 73T^{2} \) |
| 79 | \( 1 + 8.13T + 79T^{2} \) |
| 83 | \( 1 + 5.57T + 83T^{2} \) |
| 89 | \( 1 - 3.83T + 89T^{2} \) |
| 97 | \( 1 + 1.94T + 97T^{2} \) |
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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
−7.46950484495856977936906471419, −6.74445570926561938424605562017, −5.99712788204253598707028486533, −5.30611497389425129248376088195, −4.58482005864362508387109887228, −3.90378642240516445936448805422, −3.03686875319789679456363576559, −2.21393132171979476435624424493, −1.23893695926593487079762820996, 0,
1.23893695926593487079762820996, 2.21393132171979476435624424493, 3.03686875319789679456363576559, 3.90378642240516445936448805422, 4.58482005864362508387109887228, 5.30611497389425129248376088195, 5.99712788204253598707028486533, 6.74445570926561938424605562017, 7.46950484495856977936906471419
