| L(s) = 1 | + 2.14·2-s − 2.47·3-s + 2.61·4-s + 1.19·5-s − 5.31·6-s − 3.58·7-s + 1.31·8-s + 3.11·9-s + 2.57·10-s + 2.28·11-s − 6.45·12-s − 3.13·13-s − 7.68·14-s − 2.96·15-s − 2.40·16-s − 5.69·17-s + 6.69·18-s + 3.15·19-s + 3.12·20-s + 8.85·21-s + 4.91·22-s − 1.79·23-s − 3.24·24-s − 3.56·25-s − 6.72·26-s − 0.294·27-s − 9.34·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 1.42·3-s + 1.30·4-s + 0.535·5-s − 2.16·6-s − 1.35·7-s + 0.463·8-s + 1.03·9-s + 0.813·10-s + 0.689·11-s − 1.86·12-s − 0.869·13-s − 2.05·14-s − 0.764·15-s − 0.601·16-s − 1.38·17-s + 1.57·18-s + 0.724·19-s + 0.699·20-s + 1.93·21-s + 1.04·22-s − 0.374·23-s − 0.662·24-s − 0.713·25-s − 1.31·26-s − 0.0566·27-s − 1.76·28-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 | \( 1 - 2.14T + 2T^{2} \) |
| 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 - 2.28T + 11T^{2} \) |
| 13 | \( 1 + 3.13T + 13T^{2} \) |
| 17 | \( 1 + 5.69T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 + 6.64T + 29T^{2} \) |
| 31 | \( 1 + 0.158T + 31T^{2} \) |
| 37 | \( 1 + 9.23T + 37T^{2} \) |
| 41 | \( 1 - 2.91T + 41T^{2} \) |
| 43 | \( 1 + 1.84T + 43T^{2} \) |
| 47 | \( 1 - 8.88T + 47T^{2} \) |
| 53 | \( 1 + 4.07T + 53T^{2} \) |
| 59 | \( 1 - 3.88T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 + 5.15T + 67T^{2} \) |
| 71 | \( 1 + 2.03T + 71T^{2} \) |
| 73 | \( 1 - 4.11T + 73T^{2} \) |
| 79 | \( 1 - 3.82T + 79T^{2} \) |
| 83 | \( 1 - 1.03T + 83T^{2} \) |
| 89 | \( 1 - 0.990T + 89T^{2} \) |
| 97 | \( 1 - 2.18T + 97T^{2} \) |
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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
−9.744043932845911561225161773613, −9.100347801724312527531367977417, −7.12838885393641840314998081234, −6.63755721506419067669353634401, −5.90630576672125088837516111085, −5.38465940356301026350410723157, −4.40261028155242427140955170072, −3.49420827811750291790101467460, −2.20222555740424312960013576143, 0,
2.20222555740424312960013576143, 3.49420827811750291790101467460, 4.40261028155242427140955170072, 5.38465940356301026350410723157, 5.90630576672125088837516111085, 6.63755721506419067669353634401, 7.12838885393641840314998081234, 9.100347801724312527531367977417, 9.744043932845911561225161773613
