| L(s) = 1 | − 1.41·3-s − 1.41·5-s − 7-s − 0.999·9-s + 2.82·11-s + 4.24·13-s + 2.00·15-s − 6·17-s + 4.24·19-s + 1.41·21-s + 6·23-s − 2.99·25-s + 5.65·27-s − 2.82·29-s − 4·31-s − 4.00·33-s + 1.41·35-s + 8.48·37-s − 6·39-s − 6·41-s − 8.48·43-s + 1.41·45-s + 49-s + 8.48·51-s − 5.65·53-s − 4.00·55-s − 6·57-s + ⋯ |
| L(s) = 1 | − 0.816·3-s − 0.632·5-s − 0.377·7-s − 0.333·9-s + 0.852·11-s + 1.17·13-s + 0.516·15-s − 1.45·17-s + 0.973·19-s + 0.308·21-s + 1.25·23-s − 0.599·25-s + 1.08·27-s − 0.525·29-s − 0.718·31-s − 0.696·33-s + 0.239·35-s + 1.39·37-s − 0.960·39-s − 0.937·41-s − 1.29·43-s + 0.210·45-s + 0.142·49-s + 1.18·51-s − 0.777·53-s − 0.539·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.41T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.940924759521778181505379343724, −8.164877637411407172200609504582, −7.08915894283415908958162536442, −6.46212586418590935164233249435, −5.76597064740174175738918312023, −4.78778939165836108718564571290, −3.86964436952725826027560565534, −3.02895939590336338139094924453, −1.37582359136721476244828693567, 0,
1.37582359136721476244828693567, 3.02895939590336338139094924453, 3.86964436952725826027560565534, 4.78778939165836108718564571290, 5.76597064740174175738918312023, 6.46212586418590935164233249435, 7.08915894283415908958162536442, 8.164877637411407172200609504582, 8.940924759521778181505379343724
