L(s) = 1 | + 1.33·2-s − 3-s − 0.213·4-s − 2.23·5-s − 1.33·6-s + 7-s − 2.95·8-s + 9-s − 2.99·10-s + 1.45·11-s + 0.213·12-s − 2.95·13-s + 1.33·14-s + 2.23·15-s − 3.52·16-s + 2.07·17-s + 1.33·18-s + 1.00·19-s + 0.477·20-s − 21-s + 1.94·22-s + 0.269·23-s + 2.95·24-s + 0.00438·25-s − 3.94·26-s − 27-s − 0.213·28-s + ⋯ |
L(s) = 1 | + 0.945·2-s − 0.577·3-s − 0.106·4-s − 1.00·5-s − 0.545·6-s + 0.377·7-s − 1.04·8-s + 0.333·9-s − 0.945·10-s + 0.437·11-s + 0.0616·12-s − 0.818·13-s + 0.357·14-s + 0.577·15-s − 0.881·16-s + 0.504·17-s + 0.315·18-s + 0.230·19-s + 0.106·20-s − 0.218·21-s + 0.413·22-s + 0.0561·23-s + 0.603·24-s + 0.000876·25-s − 0.773·26-s − 0.192·27-s − 0.0403·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 - 1.33T + 2T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 - 1.00T + 19T^{2} \) |
| 23 | \( 1 - 0.269T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 - 9.24T + 31T^{2} \) |
| 37 | \( 1 + 1.52T + 37T^{2} \) |
| 41 | \( 1 - 0.864T + 41T^{2} \) |
| 43 | \( 1 + 8.66T + 43T^{2} \) |
| 47 | \( 1 - 6.60T + 47T^{2} \) |
| 53 | \( 1 + 2.69T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 0.682T + 61T^{2} \) |
| 67 | \( 1 + 0.953T + 67T^{2} \) |
| 71 | \( 1 - 5.11T + 71T^{2} \) |
| 73 | \( 1 + 3.39T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 6.82T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 3.72T + 97T^{2} \) |
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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
−7.45759497322909549951679756895, −6.57685634404268778469880798030, −6.05620721339016033102345979624, −5.02194102048596707552070117250, −4.81751636420939977646123094889, −4.03114502697181661429045213743, −3.40361714343582264141936031575, −2.51360634720526538288297811572, −1.09236853352556820434116110608, 0,
1.09236853352556820434116110608, 2.51360634720526538288297811572, 3.40361714343582264141936031575, 4.03114502697181661429045213743, 4.81751636420939977646123094889, 5.02194102048596707552070117250, 6.05620721339016033102345979624, 6.57685634404268778469880798030, 7.45759497322909549951679756895