| L(s) = 1 | − 3-s − 2.57·5-s + 1.34·7-s + 9-s − 2.43·11-s + 3.37·13-s + 2.57·15-s − 6.14·17-s + 3.44·19-s − 1.34·21-s − 0.344·23-s + 1.61·25-s − 27-s + 1.68·29-s + 6.62·31-s + 2.43·33-s − 3.45·35-s + 2.45·37-s − 3.37·39-s + 8.88·41-s − 1.86·43-s − 2.57·45-s − 11.0·47-s − 5.19·49-s + 6.14·51-s − 8.49·53-s + 6.26·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.14·5-s + 0.508·7-s + 0.333·9-s − 0.734·11-s + 0.936·13-s + 0.663·15-s − 1.49·17-s + 0.791·19-s − 0.293·21-s − 0.0718·23-s + 0.322·25-s − 0.192·27-s + 0.313·29-s + 1.18·31-s + 0.423·33-s − 0.584·35-s + 0.403·37-s − 0.540·39-s + 1.38·41-s − 0.284·43-s − 0.383·45-s − 1.60·47-s − 0.741·49-s + 0.861·51-s − 1.16·53-s + 0.844·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 2.57T + 5T^{2} \) |
| 7 | \( 1 - 1.34T + 7T^{2} \) |
| 11 | \( 1 + 2.43T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 + 6.14T + 17T^{2} \) |
| 19 | \( 1 - 3.44T + 19T^{2} \) |
| 23 | \( 1 + 0.344T + 23T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 - 6.62T + 31T^{2} \) |
| 37 | \( 1 - 2.45T + 37T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 + 1.86T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 8.49T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 14.9T + 61T^{2} \) |
| 67 | \( 1 + 0.0941T + 67T^{2} \) |
| 71 | \( 1 + 6.63T + 71T^{2} \) |
| 73 | \( 1 + 7.26T + 73T^{2} \) |
| 79 | \( 1 + 7.55T + 79T^{2} \) |
| 83 | \( 1 + 9.70T + 83T^{2} \) |
| 89 | \( 1 - 9.77T + 89T^{2} \) |
| 97 | \( 1 + 7.10T + 97T^{2} \) |
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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
−8.176274341598224837840249373032, −7.38762163628192880670769097575, −6.63991512089716889498747664508, −5.88776558286473271712040808312, −4.89943329596634185970882165228, −4.40555382372022425195280356622, −3.56237467931551269863850639720, −2.51337817973767537309853150524, −1.19215116960988816790743645844, 0,
1.19215116960988816790743645844, 2.51337817973767537309853150524, 3.56237467931551269863850639720, 4.40555382372022425195280356622, 4.89943329596634185970882165228, 5.88776558286473271712040808312, 6.63991512089716889498747664508, 7.38762163628192880670769097575, 8.176274341598224837840249373032
