| L(s) = 1 | − 1.41·5-s − 3.16·7-s − 4.47·11-s + 4.47·13-s + 6.32·17-s + 2.82·19-s − 4·23-s − 2.99·25-s + 4.24·29-s + 3.16·31-s + 4.47·35-s + 4.47·37-s + 6.32·41-s + 8.48·43-s − 12·47-s + 3.00·49-s − 7.07·53-s + 6.32·55-s − 13.4·61-s − 6.32·65-s − 8·71-s − 4·73-s + 14.1·77-s − 3.16·79-s + 4.47·83-s − 8.94·85-s − 14.1·91-s + ⋯ |
| L(s) = 1 | − 0.632·5-s − 1.19·7-s − 1.34·11-s + 1.24·13-s + 1.53·17-s + 0.648·19-s − 0.834·23-s − 0.599·25-s + 0.787·29-s + 0.567·31-s + 0.755·35-s + 0.735·37-s + 0.987·41-s + 1.29·43-s − 1.75·47-s + 0.428·49-s − 0.971·53-s + 0.852·55-s − 1.71·61-s − 0.784·65-s − 0.949·71-s − 0.468·73-s + 1.61·77-s − 0.355·79-s + 0.490·83-s − 0.970·85-s − 1.48·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 3.16T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 6.32T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 3.16T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.71901682963452626702860250759, −7.61882369925751808662182739051, −6.19197367857826944517418669308, −6.04648189130571719532733359649, −5.01903926186814254744349846222, −4.02486312705886929116497459477, −3.28043288777748724398759137488, −2.74666756087266170182928605034, −1.19653150854888074006058070817, 0,
1.19653150854888074006058070817, 2.74666756087266170182928605034, 3.28043288777748724398759137488, 4.02486312705886929116497459477, 5.01903926186814254744349846222, 6.04648189130571719532733359649, 6.19197367857826944517418669308, 7.61882369925751808662182739051, 7.71901682963452626702860250759
