| L(s) = 1 | − 0.0778·3-s + 1.70·5-s + 3.05·7-s − 2.99·9-s − 2.72·11-s − 0.571·13-s − 0.133·15-s − 2.03·17-s − 1.65·19-s − 0.237·21-s + 1.92·23-s − 2.07·25-s + 0.466·27-s − 0.834·29-s − 6.56·31-s + 0.212·33-s + 5.22·35-s − 8.02·37-s + 0.0445·39-s − 2.26·41-s + 7.86·43-s − 5.11·45-s − 1.38·47-s + 2.33·49-s + 0.158·51-s − 10.7·53-s − 4.65·55-s + ⋯ |
| L(s) = 1 | − 0.0449·3-s + 0.764·5-s + 1.15·7-s − 0.997·9-s − 0.821·11-s − 0.158·13-s − 0.0343·15-s − 0.492·17-s − 0.380·19-s − 0.0519·21-s + 0.401·23-s − 0.415·25-s + 0.0898·27-s − 0.155·29-s − 1.17·31-s + 0.0369·33-s + 0.882·35-s − 1.31·37-s + 0.00713·39-s − 0.353·41-s + 1.19·43-s − 0.762·45-s − 0.202·47-s + 0.333·49-s + 0.0221·51-s − 1.47·53-s − 0.628·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 0.0778T + 3T^{2} \) |
| 5 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 + 0.571T + 13T^{2} \) |
| 17 | \( 1 + 2.03T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 - 1.92T + 23T^{2} \) |
| 29 | \( 1 + 0.834T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 + 8.02T + 37T^{2} \) |
| 41 | \( 1 + 2.26T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 3.74T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 4.23T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 + 3.47T + 73T^{2} \) |
| 79 | \( 1 - 8.59T + 79T^{2} \) |
| 83 | \( 1 - 7.83T + 83T^{2} \) |
| 89 | \( 1 + 0.239T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−8.064757374263820095324799027647, −7.51193870002388565784779565099, −6.51527054498819737652479013740, −5.67461256923284830004432931148, −5.22555358543555339270649654665, −4.47201574957897912147158034335, −3.27231403610749200434011371944, −2.31746350888857263957991276420, −1.65290158108869069712290694451, 0,
1.65290158108869069712290694451, 2.31746350888857263957991276420, 3.27231403610749200434011371944, 4.47201574957897912147158034335, 5.22555358543555339270649654665, 5.67461256923284830004432931148, 6.51527054498819737652479013740, 7.51193870002388565784779565099, 8.064757374263820095324799027647
