| L(s) = 1 | − 1.21·2-s + 3-s − 0.532·4-s + 2.05·5-s − 1.21·6-s − 4.52·7-s + 3.06·8-s + 9-s − 2.49·10-s + 0.558·11-s − 0.532·12-s + 13-s + 5.47·14-s + 2.05·15-s − 2.65·16-s + 3.74·17-s − 1.21·18-s + 4.79·19-s − 1.09·20-s − 4.52·21-s − 0.676·22-s − 7.25·23-s + 3.06·24-s − 0.767·25-s − 1.21·26-s + 27-s + 2.40·28-s + ⋯ |
| L(s) = 1 | − 0.856·2-s + 0.577·3-s − 0.266·4-s + 0.920·5-s − 0.494·6-s − 1.70·7-s + 1.08·8-s + 0.333·9-s − 0.788·10-s + 0.168·11-s − 0.153·12-s + 0.277·13-s + 1.46·14-s + 0.531·15-s − 0.663·16-s + 0.907·17-s − 0.285·18-s + 1.09·19-s − 0.244·20-s − 0.986·21-s − 0.144·22-s − 1.51·23-s + 0.626·24-s − 0.153·25-s − 0.237·26-s + 0.192·27-s + 0.454·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 5 | \( 1 - 2.05T + 5T^{2} \) |
| 7 | \( 1 + 4.52T + 7T^{2} \) |
| 11 | \( 1 - 0.558T + 11T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 23 | \( 1 + 7.25T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 1.34T + 31T^{2} \) |
| 37 | \( 1 + 3.13T + 37T^{2} \) |
| 41 | \( 1 - 1.22T + 41T^{2} \) |
| 43 | \( 1 + 1.50T + 43T^{2} \) |
| 47 | \( 1 + 4.16T + 47T^{2} \) |
| 53 | \( 1 + 3.69T + 53T^{2} \) |
| 59 | \( 1 - 4.12T + 59T^{2} \) |
| 61 | \( 1 - 8.91T + 61T^{2} \) |
| 67 | \( 1 - 0.875T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 - 7.66T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 5.54T + 83T^{2} \) |
| 89 | \( 1 + 6.04T + 89T^{2} \) |
| 97 | \( 1 - 2.26T + 97T^{2} \) |
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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
−8.180813489266921919800470308535, −7.50315034512014111186940181565, −6.77630520757720324321013988275, −5.87759624720563322384154423862, −5.34194215085804949067967536519, −3.86656637630539077746805088520, −3.49034420543957826430735019450, −2.29686737943602124830692796440, −1.35328106629548867043522882601, 0,
1.35328106629548867043522882601, 2.29686737943602124830692796440, 3.49034420543957826430735019450, 3.86656637630539077746805088520, 5.34194215085804949067967536519, 5.87759624720563322384154423862, 6.77630520757720324321013988275, 7.50315034512014111186940181565, 8.180813489266921919800470308535
