| L(s) = 1 | − 1.86·2-s + 1.46·4-s + 5-s + 2.63·7-s + 0.992·8-s − 1.86·10-s − 6.48·13-s − 4.89·14-s − 4.78·16-s + 4.72·17-s + 0.532·19-s + 1.46·20-s + 4.54·23-s + 25-s + 12.0·26-s + 3.86·28-s − 10.6·29-s + 0.190·31-s + 6.91·32-s − 8.79·34-s + 2.63·35-s − 4.22·37-s − 0.992·38-s + 0.992·40-s + 0.605·41-s − 9.24·43-s − 8.46·46-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.733·4-s + 0.447·5-s + 0.994·7-s + 0.350·8-s − 0.588·10-s − 1.79·13-s − 1.30·14-s − 1.19·16-s + 1.14·17-s + 0.122·19-s + 0.328·20-s + 0.947·23-s + 0.200·25-s + 2.36·26-s + 0.729·28-s − 1.97·29-s + 0.0342·31-s + 1.22·32-s − 1.50·34-s + 0.444·35-s − 0.695·37-s − 0.160·38-s + 0.156·40-s + 0.0945·41-s − 1.41·43-s − 1.24·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 7 | \( 1 - 2.63T + 7T^{2} \) |
| 13 | \( 1 + 6.48T + 13T^{2} \) |
| 17 | \( 1 - 4.72T + 17T^{2} \) |
| 19 | \( 1 - 0.532T + 19T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 - 0.190T + 31T^{2} \) |
| 37 | \( 1 + 4.22T + 37T^{2} \) |
| 41 | \( 1 - 0.605T + 41T^{2} \) |
| 43 | \( 1 + 9.24T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 + 6.69T + 53T^{2} \) |
| 59 | \( 1 - 5.92T + 59T^{2} \) |
| 61 | \( 1 + 3.21T + 61T^{2} \) |
| 67 | \( 1 - 5.22T + 67T^{2} \) |
| 71 | \( 1 - 2.74T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 1.21T + 79T^{2} \) |
| 83 | \( 1 + 4.66T + 83T^{2} \) |
| 89 | \( 1 + 6.40T + 89T^{2} \) |
| 97 | \( 1 + 7.32T + 97T^{2} \) |
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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
−7.80607834049894512300859087228, −7.39497200715791249433988186902, −6.79816692849537792220966782547, −5.41556908564600917915597169946, −5.15541521693433450814508751034, −4.19771998921655499549499802982, −2.92675077876668244009316078931, −1.97196191777892256130989503777, −1.30889461772482123013438762948, 0,
1.30889461772482123013438762948, 1.97196191777892256130989503777, 2.92675077876668244009316078931, 4.19771998921655499549499802982, 5.15541521693433450814508751034, 5.41556908564600917915597169946, 6.79816692849537792220966782547, 7.39497200715791249433988186902, 7.80607834049894512300859087228
