| L(s) = 1 | + 0.470·2-s − 1.77·4-s − 3.24·5-s − 7-s − 1.77·8-s − 1.52·10-s − 2.47·13-s − 0.470·14-s + 2.71·16-s + 3.24·17-s − 5.30·19-s + 5.77·20-s + 8.86·23-s + 5.55·25-s − 1.16·26-s + 1.77·28-s + 2.47·29-s − 6.49·31-s + 4.83·32-s + 1.52·34-s + 3.24·35-s + 1.77·37-s − 2.49·38-s + 5.77·40-s + 1.28·41-s − 4.89·43-s + 4.17·46-s + ⋯ |
| L(s) = 1 | + 0.332·2-s − 0.889·4-s − 1.45·5-s − 0.377·7-s − 0.628·8-s − 0.483·10-s − 0.685·13-s − 0.125·14-s + 0.679·16-s + 0.788·17-s − 1.21·19-s + 1.29·20-s + 1.84·23-s + 1.11·25-s − 0.228·26-s + 0.336·28-s + 0.458·29-s − 1.16·31-s + 0.855·32-s + 0.262·34-s + 0.549·35-s + 0.292·37-s − 0.405·38-s + 0.913·40-s + 0.199·41-s − 0.746·43-s + 0.615·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 5 | \( 1 + 3.24T + 5T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 - 3.24T + 17T^{2} \) |
| 19 | \( 1 + 5.30T + 19T^{2} \) |
| 23 | \( 1 - 8.86T + 23T^{2} \) |
| 29 | \( 1 - 2.47T + 29T^{2} \) |
| 31 | \( 1 + 6.49T + 31T^{2} \) |
| 37 | \( 1 - 1.77T + 37T^{2} \) |
| 41 | \( 1 - 1.28T + 41T^{2} \) |
| 43 | \( 1 + 4.89T + 43T^{2} \) |
| 47 | \( 1 + 3.96T + 47T^{2} \) |
| 53 | \( 1 - 9.77T + 53T^{2} \) |
| 59 | \( 1 + 2.41T + 59T^{2} \) |
| 61 | \( 1 - 4.74T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 3.19T + 71T^{2} \) |
| 73 | \( 1 - 8.98T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 + 7.86T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−7.44925753804600215320038442780, −7.02018855868030406447974577884, −6.08730902441472551775736122778, −5.10661737654976694723737557982, −4.74493085311436246762915935773, −3.80481081887071681648686194081, −3.48287112928752498609953364640, −2.53206740589611133411818548199, −0.906686163267401990341140566352, 0,
0.906686163267401990341140566352, 2.53206740589611133411818548199, 3.48287112928752498609953364640, 3.80481081887071681648686194081, 4.74493085311436246762915935773, 5.10661737654976694723737557982, 6.08730902441472551775736122778, 7.02018855868030406447974577884, 7.44925753804600215320038442780
