| L(s) = 1 | + 0.413·2-s + 0.397·3-s − 1.82·4-s + 1.10·5-s + 0.164·6-s − 2.64·7-s − 1.58·8-s − 2.84·9-s + 0.455·10-s + 6.27·11-s − 0.726·12-s + 0.831·13-s − 1.09·14-s + 0.437·15-s + 3.00·16-s − 4.61·17-s − 1.17·18-s − 1.24·19-s − 2.01·20-s − 1.05·21-s + 2.59·22-s − 7.02·23-s − 0.629·24-s − 3.78·25-s + 0.343·26-s − 2.32·27-s + 4.83·28-s + ⋯ |
| L(s) = 1 | + 0.292·2-s + 0.229·3-s − 0.914·4-s + 0.492·5-s + 0.0671·6-s − 0.999·7-s − 0.560·8-s − 0.947·9-s + 0.144·10-s + 1.89·11-s − 0.209·12-s + 0.230·13-s − 0.292·14-s + 0.113·15-s + 0.750·16-s − 1.12·17-s − 0.277·18-s − 0.286·19-s − 0.450·20-s − 0.229·21-s + 0.553·22-s − 1.46·23-s − 0.128·24-s − 0.757·25-s + 0.0674·26-s − 0.446·27-s + 0.914·28-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 | \( 1 - 0.413T + 2T^{2} \) |
| 3 | \( 1 - 0.397T + 3T^{2} \) |
| 5 | \( 1 - 1.10T + 5T^{2} \) |
| 7 | \( 1 + 2.64T + 7T^{2} \) |
| 11 | \( 1 - 6.27T + 11T^{2} \) |
| 13 | \( 1 - 0.831T + 13T^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 + 7.02T + 23T^{2} \) |
| 29 | \( 1 + 1.10T + 29T^{2} \) |
| 31 | \( 1 + 4.26T + 31T^{2} \) |
| 37 | \( 1 + 5.53T + 37T^{2} \) |
| 41 | \( 1 + 2.46T + 41T^{2} \) |
| 43 | \( 1 + 8.86T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 + 2.41T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 3.24T + 71T^{2} \) |
| 73 | \( 1 + 0.360T + 73T^{2} \) |
| 79 | \( 1 - 6.60T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 1.35T + 97T^{2} \) |
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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
−9.497789939191826805662196511682, −8.870976742108720903918263220294, −8.254039877394263668164585908109, −6.62528162111629380755246236867, −6.24250339727380370754089081338, −5.26371919352259651687441750770, −3.93223412412121002870240298753, −3.51500908417188432971149738316, −1.96656855933845988131539869926, 0,
1.96656855933845988131539869926, 3.51500908417188432971149738316, 3.93223412412121002870240298753, 5.26371919352259651687441750770, 6.24250339727380370754089081338, 6.62528162111629380755246236867, 8.254039877394263668164585908109, 8.870976742108720903918263220294, 9.497789939191826805662196511682
