| L(s) = 1 | + 2.32·2-s + 3.42·4-s + 3.16·5-s − 7-s + 3.32·8-s + 7.38·10-s − 0.406·11-s + 3.07·13-s − 2.32·14-s + 0.886·16-s − 17-s − 1.57·19-s + 10.8·20-s − 0.946·22-s − 0.852·23-s + 5.04·25-s + 7.17·26-s − 3.42·28-s + 1.49·29-s − 0.316·31-s − 4.57·32-s − 2.32·34-s − 3.16·35-s + 10.6·37-s − 3.67·38-s + 10.5·40-s − 11.8·41-s + ⋯ |
| L(s) = 1 | + 1.64·2-s + 1.71·4-s + 1.41·5-s − 0.377·7-s + 1.17·8-s + 2.33·10-s − 0.122·11-s + 0.854·13-s − 0.622·14-s + 0.221·16-s − 0.242·17-s − 0.362·19-s + 2.42·20-s − 0.201·22-s − 0.177·23-s + 1.00·25-s + 1.40·26-s − 0.647·28-s + 0.278·29-s − 0.0568·31-s − 0.809·32-s − 0.399·34-s − 0.535·35-s + 1.74·37-s − 0.596·38-s + 1.66·40-s − 1.85·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.868695574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.868695574\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 2.32T + 2T^{2} \) |
| 5 | \( 1 - 3.16T + 5T^{2} \) |
| 11 | \( 1 + 0.406T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 19 | \( 1 + 1.57T + 19T^{2} \) |
| 23 | \( 1 + 0.852T + 23T^{2} \) |
| 29 | \( 1 - 1.49T + 29T^{2} \) |
| 31 | \( 1 + 0.316T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + 9.09T + 47T^{2} \) |
| 53 | \( 1 - 4.34T + 53T^{2} \) |
| 59 | \( 1 + 7.15T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 - 5.48T + 67T^{2} \) |
| 71 | \( 1 - 8.73T + 71T^{2} \) |
| 73 | \( 1 - 6.25T + 73T^{2} \) |
| 79 | \( 1 - 3.77T + 79T^{2} \) |
| 83 | \( 1 - 5.35T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 0.623T + 97T^{2} \) |
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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
−10.01617372088883412442866584375, −9.232780643362219863575807214366, −8.166561169304786587105265140756, −6.69042861127185264420172008974, −6.34236019401565509210068832056, −5.56445135232467930104627945510, −4.79019524407272346299057270507, −3.70338840549253129316472031337, −2.73885673412750403632093243225, −1.75105383020301437125779422172,
1.75105383020301437125779422172, 2.73885673412750403632093243225, 3.70338840549253129316472031337, 4.79019524407272346299057270507, 5.56445135232467930104627945510, 6.34236019401565509210068832056, 6.69042861127185264420172008974, 8.166561169304786587105265140756, 9.232780643362219863575807214366, 10.01617372088883412442866584375
