| L(s) = 1 | + 1.30·2-s − 1.71·3-s − 0.296·4-s + 2.67·5-s − 2.24·6-s + 1.19·7-s − 2.99·8-s − 0.0520·9-s + 3.49·10-s − 3.99·11-s + 0.509·12-s − 5.32·13-s + 1.55·14-s − 4.59·15-s − 3.31·16-s + 1.44·17-s − 0.0678·18-s + 0.225·19-s − 0.794·20-s − 2.04·21-s − 5.21·22-s + 1.91·23-s + 5.14·24-s + 2.16·25-s − 6.94·26-s + 5.24·27-s − 0.353·28-s + ⋯ |
| L(s) = 1 | + 0.922·2-s − 0.991·3-s − 0.148·4-s + 1.19·5-s − 0.914·6-s + 0.450·7-s − 1.05·8-s − 0.0173·9-s + 1.10·10-s − 1.20·11-s + 0.147·12-s − 1.47·13-s + 0.415·14-s − 1.18·15-s − 0.829·16-s + 0.349·17-s − 0.0159·18-s + 0.0516·19-s − 0.177·20-s − 0.446·21-s − 1.11·22-s + 0.399·23-s + 1.05·24-s + 0.433·25-s − 1.36·26-s + 1.00·27-s − 0.0668·28-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 | \( 1 - 1.30T + 2T^{2} \) |
| 3 | \( 1 + 1.71T + 3T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 3.99T + 11T^{2} \) |
| 13 | \( 1 + 5.32T + 13T^{2} \) |
| 17 | \( 1 - 1.44T + 17T^{2} \) |
| 19 | \( 1 - 0.225T + 19T^{2} \) |
| 23 | \( 1 - 1.91T + 23T^{2} \) |
| 29 | \( 1 + 7.90T + 29T^{2} \) |
| 31 | \( 1 + 0.155T + 31T^{2} \) |
| 37 | \( 1 + 2.55T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 1.40T + 43T^{2} \) |
| 47 | \( 1 + 7.63T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 3.42T + 59T^{2} \) |
| 61 | \( 1 + 8.98T + 61T^{2} \) |
| 67 | \( 1 - 6.23T + 67T^{2} \) |
| 71 | \( 1 - 0.117T + 71T^{2} \) |
| 73 | \( 1 + 5.48T + 73T^{2} \) |
| 79 | \( 1 - 5.03T + 79T^{2} \) |
| 83 | \( 1 - 3.96T + 83T^{2} \) |
| 89 | \( 1 - 8.61T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−9.824591676285854309358317876997, −8.896635175249430756693475133466, −7.75110134748855051033377908742, −6.65159334344276775887546965868, −5.70073626250434910505505416931, −5.21071414639228158053325365478, −4.81085807860940694056850313931, −3.17828426164196721303691706395, −2.09290204262615585234390924091, 0,
2.09290204262615585234390924091, 3.17828426164196721303691706395, 4.81085807860940694056850313931, 5.21071414639228158053325365478, 5.70073626250434910505505416931, 6.65159334344276775887546965868, 7.75110134748855051033377908742, 8.896635175249430756693475133466, 9.824591676285854309358317876997
