| L(s) = 1 | + 1.28·2-s − 0.345·4-s − 0.612·5-s + 1.03·7-s − 3.01·8-s − 0.788·10-s − 0.227·11-s + 3.38·13-s + 1.32·14-s − 3.18·16-s − 7.12·17-s + 3.40·19-s + 0.211·20-s − 0.293·22-s − 6.91·23-s − 4.62·25-s + 4.34·26-s − 0.356·28-s − 0.569·29-s + 4.93·31-s + 1.93·32-s − 9.16·34-s − 0.631·35-s − 5.37·37-s + 4.37·38-s + 1.84·40-s − 10.7·41-s + ⋯ |
| L(s) = 1 | + 0.909·2-s − 0.172·4-s − 0.274·5-s + 0.389·7-s − 1.06·8-s − 0.249·10-s − 0.0687·11-s + 0.937·13-s + 0.354·14-s − 0.797·16-s − 1.72·17-s + 0.780·19-s + 0.0473·20-s − 0.0625·22-s − 1.44·23-s − 0.924·25-s + 0.852·26-s − 0.0672·28-s − 0.105·29-s + 0.886·31-s + 0.341·32-s − 1.57·34-s − 0.106·35-s − 0.884·37-s + 0.709·38-s + 0.292·40-s − 1.67·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 1.28T + 2T^{2} \) |
| 5 | \( 1 + 0.612T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 + 0.227T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 + 7.12T + 17T^{2} \) |
| 19 | \( 1 - 3.40T + 19T^{2} \) |
| 23 | \( 1 + 6.91T + 23T^{2} \) |
| 29 | \( 1 + 0.569T + 29T^{2} \) |
| 31 | \( 1 - 4.93T + 31T^{2} \) |
| 37 | \( 1 + 5.37T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 + 0.910T + 43T^{2} \) |
| 47 | \( 1 - 8.50T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 3.65T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 9.48T + 71T^{2} \) |
| 73 | \( 1 + 7.10T + 73T^{2} \) |
| 79 | \( 1 - 0.366T + 79T^{2} \) |
| 83 | \( 1 + 17.8T + 83T^{2} \) |
| 89 | \( 1 - 7.54T + 89T^{2} \) |
| 97 | \( 1 + 7.85T + 97T^{2} \) |
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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
−8.596058668850017656846750007783, −8.053084387109213955388019597609, −6.91907190918899566976725120464, −6.13463102676438561857948769300, −5.43611787715243087696437807673, −4.46302723556472528533286019730, −3.98054768671116740971709307047, −3.01753943087213503124201685564, −1.78471136581592608713704969146, 0,
1.78471136581592608713704969146, 3.01753943087213503124201685564, 3.98054768671116740971709307047, 4.46302723556472528533286019730, 5.43611787715243087696437807673, 6.13463102676438561857948769300, 6.91907190918899566976725120464, 8.053084387109213955388019597609, 8.596058668850017656846750007783
