L(s) = 1 | + 2.10·2-s + 3-s + 2.45·4-s − 3.12·5-s + 2.10·6-s − 7-s + 0.951·8-s + 9-s − 6.58·10-s − 3.21·11-s + 2.45·12-s + 2.86·13-s − 2.10·14-s − 3.12·15-s − 2.89·16-s + 3.21·17-s + 2.10·18-s + 5.92·19-s − 7.64·20-s − 21-s − 6.79·22-s + 7.17·23-s + 0.951·24-s + 4.73·25-s + 6.05·26-s + 27-s − 2.45·28-s + ⋯ |
L(s) = 1 | + 1.49·2-s + 0.577·3-s + 1.22·4-s − 1.39·5-s + 0.861·6-s − 0.377·7-s + 0.336·8-s + 0.333·9-s − 2.08·10-s − 0.970·11-s + 0.707·12-s + 0.795·13-s − 0.563·14-s − 0.805·15-s − 0.723·16-s + 0.780·17-s + 0.497·18-s + 1.35·19-s − 1.71·20-s − 0.218·21-s − 1.44·22-s + 1.49·23-s + 0.194·24-s + 0.947·25-s + 1.18·26-s + 0.192·27-s − 0.463·28-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 | \( 1 - 2.10T + 2T^{2} \) |
| 5 | \( 1 + 3.12T + 5T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 13 | \( 1 - 2.86T + 13T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 23 | \( 1 - 7.17T + 23T^{2} \) |
| 29 | \( 1 + 4.81T + 29T^{2} \) |
| 31 | \( 1 + 8.09T + 31T^{2} \) |
| 37 | \( 1 - 0.455T + 37T^{2} \) |
| 41 | \( 1 + 3.54T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 0.893T + 47T^{2} \) |
| 53 | \( 1 + 4.91T + 53T^{2} \) |
| 59 | \( 1 + 2.42T + 59T^{2} \) |
| 61 | \( 1 + 9.72T + 61T^{2} \) |
| 67 | \( 1 + 5.54T + 67T^{2} \) |
| 71 | \( 1 - 1.95T + 71T^{2} \) |
| 73 | \( 1 + 8.59T + 73T^{2} \) |
| 79 | \( 1 + 4.56T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 4.09T + 89T^{2} \) |
| 97 | \( 1 - 9.26T + 97T^{2} \) |
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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
−7.39182450918493193409409934149, −6.88887106155956839064851223407, −5.82926501415368494807255993228, −5.23442980934624689403970781762, −4.61245696722254155003239440022, −3.69043709464847716345161893184, −3.29634592727841714453191292007, −2.93073794027780334797700971453, −1.50766628313650805226592371733, 0,
1.50766628313650805226592371733, 2.93073794027780334797700971453, 3.29634592727841714453191292007, 3.69043709464847716345161893184, 4.61245696722254155003239440022, 5.23442980934624689403970781762, 5.82926501415368494807255993228, 6.88887106155956839064851223407, 7.39182450918493193409409934149