| L(s) = 1 | − 2.44·2-s + 3.97·4-s + 2.87·5-s + 1.56·7-s − 4.82·8-s − 7.02·10-s − 6.34·11-s + 0.803·13-s − 3.81·14-s + 3.83·16-s + 3.25·19-s + 11.4·20-s + 15.5·22-s − 6.17·23-s + 3.25·25-s − 1.96·26-s + 6.20·28-s − 1.52·29-s − 2.17·31-s + 0.263·32-s + 4.48·35-s − 0.636·37-s − 7.94·38-s − 13.8·40-s + 5.61·41-s − 11.0·43-s − 25.1·44-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.98·4-s + 1.28·5-s + 0.590·7-s − 1.70·8-s − 2.22·10-s − 1.91·11-s + 0.222·13-s − 1.02·14-s + 0.959·16-s + 0.745·19-s + 2.55·20-s + 3.30·22-s − 1.28·23-s + 0.651·25-s − 0.385·26-s + 1.17·28-s − 0.283·29-s − 0.390·31-s + 0.0465·32-s + 0.758·35-s − 0.104·37-s − 1.28·38-s − 2.19·40-s + 0.877·41-s − 1.68·43-s − 3.79·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 - 0.803T + 13T^{2} \) |
| 19 | \( 1 - 3.25T + 19T^{2} \) |
| 23 | \( 1 + 6.17T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 + 2.17T + 31T^{2} \) |
| 37 | \( 1 + 0.636T + 37T^{2} \) |
| 41 | \( 1 - 5.61T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 7.46T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 + 4.03T + 59T^{2} \) |
| 61 | \( 1 + 8.36T + 61T^{2} \) |
| 67 | \( 1 - 7.75T + 67T^{2} \) |
| 71 | \( 1 + 4.51T + 71T^{2} \) |
| 73 | \( 1 + 8.28T + 73T^{2} \) |
| 79 | \( 1 + 1.50T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 - 6.12T + 89T^{2} \) |
| 97 | \( 1 + 4.84T + 97T^{2} \) |
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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
−8.457111769029998390868515626566, −7.928233239540738928003118960823, −7.36075344747812633417335117435, −6.30053366180446136238484290600, −5.62363197940952287356183284369, −4.83227634469482461550653045691, −3.06267541541045915163594609253, −2.14499882028126511109749377681, −1.53542126069137570400270125614, 0,
1.53542126069137570400270125614, 2.14499882028126511109749377681, 3.06267541541045915163594609253, 4.83227634469482461550653045691, 5.62363197940952287356183284369, 6.30053366180446136238484290600, 7.36075344747812633417335117435, 7.928233239540738928003118960823, 8.457111769029998390868515626566
