| L(s) = 1 | − 2.47·3-s + 3.58·5-s − 4.75·7-s + 3.11·9-s + 11-s − 13-s − 8.87·15-s − 6.22·17-s + 5.34·19-s + 11.7·21-s − 4.83·23-s + 7.87·25-s − 0.284·27-s − 5.28·29-s − 7.35·31-s − 2.47·33-s − 17.0·35-s − 7.58·37-s + 2.47·39-s − 6.98·41-s − 7.66·43-s + 11.1·45-s − 5.17·47-s + 15.6·49-s + 15.4·51-s + 0.169·53-s + 3.58·55-s + ⋯ |
| L(s) = 1 | − 1.42·3-s + 1.60·5-s − 1.79·7-s + 1.03·9-s + 0.301·11-s − 0.277·13-s − 2.29·15-s − 1.51·17-s + 1.22·19-s + 2.56·21-s − 1.00·23-s + 1.57·25-s − 0.0546·27-s − 0.981·29-s − 1.32·31-s − 0.430·33-s − 2.88·35-s − 1.24·37-s + 0.395·39-s − 1.09·41-s − 1.16·43-s + 1.66·45-s − 0.754·47-s + 2.23·49-s + 2.15·51-s + 0.0232·53-s + 0.483·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 - 3.58T + 5T^{2} \) |
| 7 | \( 1 + 4.75T + 7T^{2} \) |
| 17 | \( 1 + 6.22T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 + 7.35T + 31T^{2} \) |
| 37 | \( 1 + 7.58T + 37T^{2} \) |
| 41 | \( 1 + 6.98T + 41T^{2} \) |
| 43 | \( 1 + 7.66T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 - 0.169T + 53T^{2} \) |
| 59 | \( 1 - 1.21T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 9.24T + 71T^{2} \) |
| 73 | \( 1 - 5.49T + 73T^{2} \) |
| 79 | \( 1 + 7.74T + 79T^{2} \) |
| 83 | \( 1 - 3.47T + 83T^{2} \) |
| 89 | \( 1 + 4.04T + 89T^{2} \) |
| 97 | \( 1 - 3.58T + 97T^{2} \) |
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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
−10.15692710813267048104231013057, −9.693900889561031490678947424626, −8.979219861232992703387159042804, −6.96834426581701963828506469785, −6.51386852917970845515961141112, −5.78549000303967251215891819484, −5.09599337592817740470398383890, −3.46968513551905324569289744366, −1.94654803949960011287000824622, 0,
1.94654803949960011287000824622, 3.46968513551905324569289744366, 5.09599337592817740470398383890, 5.78549000303967251215891819484, 6.51386852917970845515961141112, 6.96834426581701963828506469785, 8.979219861232992703387159042804, 9.693900889561031490678947424626, 10.15692710813267048104231013057
