| L(s) = 1 | − 1.35·2-s − 3-s − 0.169·4-s − 0.738·5-s + 1.35·6-s + 3.44·7-s + 2.93·8-s + 9-s + 0.998·10-s + 1.17·11-s + 0.169·12-s − 2.89·13-s − 4.65·14-s + 0.738·15-s − 3.63·16-s − 3.96·17-s − 1.35·18-s − 3.02·19-s + 0.125·20-s − 3.44·21-s − 1.58·22-s − 23-s − 2.93·24-s − 4.45·25-s + 3.92·26-s − 27-s − 0.583·28-s + ⋯ |
| L(s) = 1 | − 0.956·2-s − 0.577·3-s − 0.0848·4-s − 0.330·5-s + 0.552·6-s + 1.30·7-s + 1.03·8-s + 0.333·9-s + 0.315·10-s + 0.353·11-s + 0.0489·12-s − 0.803·13-s − 1.24·14-s + 0.190·15-s − 0.908·16-s − 0.962·17-s − 0.318·18-s − 0.694·19-s + 0.0279·20-s − 0.751·21-s − 0.338·22-s − 0.208·23-s − 0.599·24-s − 0.891·25-s + 0.769·26-s − 0.192·27-s − 0.110·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 5 | \( 1 + 0.738T + 5T^{2} \) |
| 7 | \( 1 - 3.44T + 7T^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 + 3.96T + 17T^{2} \) |
| 19 | \( 1 + 3.02T + 19T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 - 3.14T + 37T^{2} \) |
| 41 | \( 1 + 7.62T + 41T^{2} \) |
| 43 | \( 1 - 6.17T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 - 8.65T + 67T^{2} \) |
| 71 | \( 1 + 8.41T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 9.53T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.634598769449736356284874590351, −8.127718887209450035647039955800, −7.43066343402706064386873940263, −6.62247742978267834689407265875, −5.50239519245059877007188968269, −4.46099930205908849412689395057, −4.29901426325966291609856632082, −2.33153827572209381977326186675, −1.31850784801509289209605450968, 0,
1.31850784801509289209605450968, 2.33153827572209381977326186675, 4.29901426325966291609856632082, 4.46099930205908849412689395057, 5.50239519245059877007188968269, 6.62247742978267834689407265875, 7.43066343402706064386873940263, 8.127718887209450035647039955800, 8.634598769449736356284874590351
