| L(s) = 1 | − 2.72·2-s + 5.41·4-s + 1.47·5-s + 3.37·7-s − 9.29·8-s − 4.01·10-s + 3.55·11-s + 1.58·13-s − 9.19·14-s + 14.4·16-s + 3.58·19-s + 7.97·20-s − 9.68·22-s − 1.47·23-s − 2.82·25-s − 4.31·26-s + 18.2·28-s − 0.863·29-s + 3.24·31-s − 20.8·32-s + 4.97·35-s + 7.07·37-s − 9.76·38-s − 13.6·40-s + 8.58·41-s − 6.07·43-s + 19.2·44-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 2.70·4-s + 0.659·5-s + 1.27·7-s − 3.28·8-s − 1.26·10-s + 1.07·11-s + 0.439·13-s − 2.45·14-s + 3.62·16-s + 0.822·19-s + 1.78·20-s − 2.06·22-s − 0.307·23-s − 0.565·25-s − 0.846·26-s + 3.45·28-s − 0.160·29-s + 0.583·31-s − 3.68·32-s + 0.841·35-s + 1.16·37-s − 1.58·38-s − 2.16·40-s + 1.34·41-s − 0.925·43-s + 2.90·44-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
| \(L(1)\) |
\(\approx\) |
\(1.189657259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.189657259\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.72T + 2T^{2} \) |
| 5 | \( 1 - 1.47T + 5T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 3.55T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 + 1.47T + 23T^{2} \) |
| 29 | \( 1 + 0.863T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 - 8.58T + 41T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 + 7.70T + 47T^{2} \) |
| 53 | \( 1 - 2.25T + 53T^{2} \) |
| 59 | \( 1 - 8.16T + 59T^{2} \) |
| 61 | \( 1 - 1.84T + 61T^{2} \) |
| 67 | \( 1 - 3.41T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 8.02T + 73T^{2} \) |
| 79 | \( 1 + 9.68T + 79T^{2} \) |
| 83 | \( 1 - 2.25T + 83T^{2} \) |
| 89 | \( 1 + 8.82T + 89T^{2} \) |
| 97 | \( 1 - 0.448T + 97T^{2} \) |
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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
−8.930054338095869284375822748558, −8.145261944106610940213743774777, −7.75324102897168605159178704755, −6.76589428974310058080545944073, −6.15788227617100749065671709142, −5.27324693911094940764065511151, −3.88519227414036811831219762882, −2.54909865428012052520240465712, −1.66311575056303668439068123044, −1.01558788479671898946296532451,
1.01558788479671898946296532451, 1.66311575056303668439068123044, 2.54909865428012052520240465712, 3.88519227414036811831219762882, 5.27324693911094940764065511151, 6.15788227617100749065671709142, 6.76589428974310058080545944073, 7.75324102897168605159178704755, 8.145261944106610940213743774777, 8.930054338095869284375822748558
