| L(s) = 1 | + 1.49·3-s − 1.09·5-s − 0.799·7-s − 0.776·9-s − 0.838·11-s + 2.03·13-s − 1.63·15-s − 1.82·17-s + 4.30·19-s − 1.19·21-s − 0.747·23-s − 3.79·25-s − 5.63·27-s + 2.79·29-s − 0.0226·31-s − 1.25·33-s + 0.877·35-s + 1.80·37-s + 3.03·39-s + 6.90·41-s − 8.84·43-s + 0.852·45-s − 6.50·47-s − 6.36·49-s − 2.71·51-s − 8.42·53-s + 0.920·55-s + ⋯ |
| L(s) = 1 | + 0.860·3-s − 0.490·5-s − 0.302·7-s − 0.258·9-s − 0.252·11-s + 0.563·13-s − 0.422·15-s − 0.442·17-s + 0.987·19-s − 0.260·21-s − 0.155·23-s − 0.759·25-s − 1.08·27-s + 0.518·29-s − 0.00406·31-s − 0.217·33-s + 0.148·35-s + 0.296·37-s + 0.485·39-s + 1.07·41-s − 1.34·43-s + 0.127·45-s − 0.948·47-s − 0.908·49-s − 0.380·51-s − 1.15·53-s + 0.124·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 1.49T + 3T^{2} \) |
| 5 | \( 1 + 1.09T + 5T^{2} \) |
| 7 | \( 1 + 0.799T + 7T^{2} \) |
| 11 | \( 1 + 0.838T + 11T^{2} \) |
| 13 | \( 1 - 2.03T + 13T^{2} \) |
| 17 | \( 1 + 1.82T + 17T^{2} \) |
| 19 | \( 1 - 4.30T + 19T^{2} \) |
| 23 | \( 1 + 0.747T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 + 0.0226T + 31T^{2} \) |
| 37 | \( 1 - 1.80T + 37T^{2} \) |
| 41 | \( 1 - 6.90T + 41T^{2} \) |
| 43 | \( 1 + 8.84T + 43T^{2} \) |
| 47 | \( 1 + 6.50T + 47T^{2} \) |
| 53 | \( 1 + 8.42T + 53T^{2} \) |
| 59 | \( 1 + 1.81T + 59T^{2} \) |
| 61 | \( 1 + 8.08T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 1.18T + 71T^{2} \) |
| 73 | \( 1 - 3.90T + 73T^{2} \) |
| 79 | \( 1 + 8.22T + 79T^{2} \) |
| 83 | \( 1 + 5.12T + 83T^{2} \) |
| 89 | \( 1 + 9.57T + 89T^{2} \) |
| 97 | \( 1 + 0.843T + 97T^{2} \) |
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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
−8.109314652389105632531390681808, −7.60591794602363314430815027265, −6.65813034644896398681133982055, −5.92191296436868289868206188827, −5.01566479069119974691773442502, −4.05060560603059090409994439610, −3.31750602964112403830605006640, −2.69758778600117496346834699018, −1.54235452386861216014684970780, 0,
1.54235452386861216014684970780, 2.69758778600117496346834699018, 3.31750602964112403830605006640, 4.05060560603059090409994439610, 5.01566479069119974691773442502, 5.92191296436868289868206188827, 6.65813034644896398681133982055, 7.60591794602363314430815027265, 8.109314652389105632531390681808
