| L(s) = 1 | − 2.27·2-s + 3.18·4-s − 3.40·5-s + 0.928·7-s − 2.70·8-s + 7.74·10-s − 11-s − 4.04·13-s − 2.11·14-s − 0.222·16-s − 7.35·17-s + 8.26·19-s − 10.8·20-s + 2.27·22-s + 6.65·23-s + 6.57·25-s + 9.21·26-s + 2.95·28-s − 9.43·29-s + 0.00498·31-s + 5.90·32-s + 16.7·34-s − 3.15·35-s + 5.76·37-s − 18.8·38-s + 9.18·40-s − 4.34·41-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.59·4-s − 1.52·5-s + 0.350·7-s − 0.954·8-s + 2.45·10-s − 0.301·11-s − 1.12·13-s − 0.565·14-s − 0.0555·16-s − 1.78·17-s + 1.89·19-s − 2.42·20-s + 0.485·22-s + 1.38·23-s + 1.31·25-s + 1.80·26-s + 0.558·28-s − 1.75·29-s + 0.000895·31-s + 1.04·32-s + 2.87·34-s − 0.534·35-s + 0.947·37-s − 3.05·38-s + 1.45·40-s − 0.679·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2.27T + 2T^{2} \) |
| 5 | \( 1 + 3.40T + 5T^{2} \) |
| 7 | \( 1 - 0.928T + 7T^{2} \) |
| 13 | \( 1 + 4.04T + 13T^{2} \) |
| 17 | \( 1 + 7.35T + 17T^{2} \) |
| 19 | \( 1 - 8.26T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 + 9.43T + 29T^{2} \) |
| 31 | \( 1 - 0.00498T + 31T^{2} \) |
| 37 | \( 1 - 5.76T + 37T^{2} \) |
| 41 | \( 1 + 4.34T + 41T^{2} \) |
| 43 | \( 1 + 9.32T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 3.75T + 53T^{2} \) |
| 59 | \( 1 + 0.186T + 59T^{2} \) |
| 61 | \( 1 - 4.14T + 61T^{2} \) |
| 67 | \( 1 + 2.99T + 67T^{2} \) |
| 71 | \( 1 - 9.74T + 71T^{2} \) |
| 73 | \( 1 - 0.850T + 73T^{2} \) |
| 79 | \( 1 - 7.75T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 3.28T + 97T^{2} \) |
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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
−7.69919021635454882846356343946, −7.13327884238134462828505748812, −6.70342461789952127291961010244, −5.24947367782579574573682432780, −4.74249997281779051385257947987, −3.78875596010047949957628470292, −2.86891083106335760805658042414, −2.01361589662654269637908079317, −0.830644873026048864968508025483, 0,
0.830644873026048864968508025483, 2.01361589662654269637908079317, 2.86891083106335760805658042414, 3.78875596010047949957628470292, 4.74249997281779051385257947987, 5.24947367782579574573682432780, 6.70342461789952127291961010244, 7.13327884238134462828505748812, 7.69919021635454882846356343946
