| L(s) = 1 | + 2.36·2-s + 3.57·4-s − 4.36·5-s − 3.93·7-s + 3.72·8-s − 10.2·10-s − 11-s − 2.36·13-s − 9.29·14-s + 1.63·16-s − 2.21·17-s + 5.29·19-s − 15.5·20-s − 2.36·22-s − 0.784·23-s + 14.0·25-s − 5.57·26-s − 14.0·28-s − 4.42·29-s − 6.57·31-s − 3.57·32-s − 5.23·34-s + 17.1·35-s + 6.15·37-s + 12.5·38-s − 16.2·40-s − 2.57·41-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.78·4-s − 1.95·5-s − 1.48·7-s + 1.31·8-s − 3.25·10-s − 0.301·11-s − 0.654·13-s − 2.48·14-s + 0.409·16-s − 0.537·17-s + 1.21·19-s − 3.48·20-s − 0.503·22-s − 0.163·23-s + 2.80·25-s − 1.09·26-s − 2.66·28-s − 0.821·29-s − 1.18·31-s − 0.632·32-s − 0.897·34-s + 2.90·35-s + 1.01·37-s + 2.03·38-s − 2.56·40-s − 0.402·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 5 | \( 1 + 4.36T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 + 2.21T + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + 0.784T + 23T^{2} \) |
| 29 | \( 1 + 4.42T + 29T^{2} \) |
| 31 | \( 1 + 6.57T + 31T^{2} \) |
| 37 | \( 1 - 6.15T + 37T^{2} \) |
| 41 | \( 1 + 2.57T + 41T^{2} \) |
| 43 | \( 1 + 1.93T + 43T^{2} \) |
| 47 | \( 1 - 3.29T + 47T^{2} \) |
| 53 | \( 1 - 1.00T + 53T^{2} \) |
| 59 | \( 1 + 4.35T + 59T^{2} \) |
| 61 | \( 1 - 4.93T + 61T^{2} \) |
| 67 | \( 1 - 3.23T + 67T^{2} \) |
| 71 | \( 1 + 8.45T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 2.93T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 0.430T + 97T^{2} \) |
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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
−9.849067503108357581604068688128, −8.784602401479325722396875839755, −7.37659075165393621376589771016, −7.22266409687404779740161782750, −6.08452860709521797547497495365, −5.05802311771250319967623711449, −4.12415783859339145810231386477, −3.46375769696237640780060249662, −2.78366221376082730103963280331, 0,
2.78366221376082730103963280331, 3.46375769696237640780060249662, 4.12415783859339145810231386477, 5.05802311771250319967623711449, 6.08452860709521797547497495365, 7.22266409687404779740161782750, 7.37659075165393621376589771016, 8.784602401479325722396875839755, 9.849067503108357581604068688128
