| L(s) = 1 | − 1.42·3-s − 3.45·5-s − 2.96·7-s − 0.961·9-s + 1.17·11-s + 1.76·13-s + 4.92·15-s − 3.62·17-s − 0.411·19-s + 4.22·21-s + 7.25·23-s + 6.92·25-s + 5.65·27-s + 5.51·29-s + 3.97·31-s − 1.68·33-s + 10.2·35-s − 5.35·37-s − 2.51·39-s + 0.721·41-s − 1.15·43-s + 3.32·45-s + 5.14·47-s + 1.76·49-s + 5.17·51-s − 7.17·53-s − 4.06·55-s + ⋯ |
| L(s) = 1 | − 0.824·3-s − 1.54·5-s − 1.11·7-s − 0.320·9-s + 0.355·11-s + 0.488·13-s + 1.27·15-s − 0.878·17-s − 0.0942·19-s + 0.922·21-s + 1.51·23-s + 1.38·25-s + 1.08·27-s + 1.02·29-s + 0.713·31-s − 0.292·33-s + 1.72·35-s − 0.879·37-s − 0.402·39-s + 0.112·41-s − 0.176·43-s + 0.494·45-s + 0.749·47-s + 0.251·49-s + 0.724·51-s − 0.985·53-s − 0.548·55-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 | \( 1 + 1.42T + 3T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 7 | \( 1 + 2.96T + 7T^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 3.62T + 17T^{2} \) |
| 19 | \( 1 + 0.411T + 19T^{2} \) |
| 23 | \( 1 - 7.25T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 - 3.97T + 31T^{2} \) |
| 37 | \( 1 + 5.35T + 37T^{2} \) |
| 41 | \( 1 - 0.721T + 41T^{2} \) |
| 43 | \( 1 + 1.15T + 43T^{2} \) |
| 47 | \( 1 - 5.14T + 47T^{2} \) |
| 53 | \( 1 + 7.17T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 - 6.75T + 61T^{2} \) |
| 67 | \( 1 - 9.81T + 67T^{2} \) |
| 71 | \( 1 + 8.20T + 71T^{2} \) |
| 73 | \( 1 + 7.49T + 73T^{2} \) |
| 79 | \( 1 + 5.47T + 79T^{2} \) |
| 83 | \( 1 - 8.03T + 83T^{2} \) |
| 89 | \( 1 + 8.41T + 89T^{2} \) |
| 97 | \( 1 - 8.26T + 97T^{2} \) |
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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
−8.206905033476613401863669148958, −7.01848666896690187719117621981, −6.75768019901144044688086451750, −5.99437959220763083534774138982, −4.98490774291534423848886986612, −4.29444651181435667301177215643, −3.43334478112417892376635122469, −2.79569335219899066369078428277, −0.920858679290003401890488966094, 0,
0.920858679290003401890488966094, 2.79569335219899066369078428277, 3.43334478112417892376635122469, 4.29444651181435667301177215643, 4.98490774291534423848886986612, 5.99437959220763083534774138982, 6.75768019901144044688086451750, 7.01848666896690187719117621981, 8.206905033476613401863669148958
