| L(s) = 1 | + 2.50·2-s − 1.04·3-s + 4.29·4-s − 2.58·5-s − 2.62·6-s − 2.53·7-s + 5.75·8-s − 1.90·9-s − 6.47·10-s − 5.05·11-s − 4.50·12-s − 1.86·13-s − 6.36·14-s + 2.70·15-s + 5.85·16-s + 5.69·17-s − 4.77·18-s − 1.68·19-s − 11.0·20-s + 2.65·21-s − 12.6·22-s − 0.00429·23-s − 6.03·24-s + 1.66·25-s − 4.66·26-s + 5.13·27-s − 10.8·28-s + ⋯ |
| L(s) = 1 | + 1.77·2-s − 0.605·3-s + 2.14·4-s − 1.15·5-s − 1.07·6-s − 0.958·7-s + 2.03·8-s − 0.633·9-s − 2.04·10-s − 1.52·11-s − 1.29·12-s − 0.516·13-s − 1.69·14-s + 0.698·15-s + 1.46·16-s + 1.38·17-s − 1.12·18-s − 0.386·19-s − 2.47·20-s + 0.579·21-s − 2.70·22-s − 0.000895·23-s − 1.23·24-s + 0.333·25-s − 0.915·26-s + 0.988·27-s − 2.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 983 | \( 1 + T \) |
| good | 2 | \( 1 - 2.50T + 2T^{2} \) |
| 3 | \( 1 + 1.04T + 3T^{2} \) |
| 5 | \( 1 + 2.58T + 5T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 11 | \( 1 + 5.05T + 11T^{2} \) |
| 13 | \( 1 + 1.86T + 13T^{2} \) |
| 17 | \( 1 - 5.69T + 17T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 23 | \( 1 + 0.00429T + 23T^{2} \) |
| 29 | \( 1 + 1.67T + 29T^{2} \) |
| 31 | \( 1 - 0.252T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + 7.79T + 41T^{2} \) |
| 43 | \( 1 - 6.90T + 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 - 0.466T + 53T^{2} \) |
| 59 | \( 1 - 5.24T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 5.31T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 8.27T + 79T^{2} \) |
| 83 | \( 1 + 2.02T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 3.32T + 97T^{2} \) |
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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
−10.05278824233117101933816670674, −8.444628407419057872110950368781, −7.50034392563152294746583734567, −6.84386556380672139782946945973, −5.63630995054824671839721349871, −5.39407947107423316165404174316, −4.24338343858899210664082496521, −3.32045493524723209064901450564, −2.65202796590368707623305271386, 0,
2.65202796590368707623305271386, 3.32045493524723209064901450564, 4.24338343858899210664082496521, 5.39407947107423316165404174316, 5.63630995054824671839721349871, 6.84386556380672139782946945973, 7.50034392563152294746583734567, 8.444628407419057872110950368781, 10.05278824233117101933816670674
