| L(s) = 1 | + 2.72·2-s + 5.40·4-s − 0.469·5-s + 1.03·7-s + 9.28·8-s − 1.27·10-s + 1.84·11-s − 0.700·13-s + 2.82·14-s + 14.4·16-s − 0.793·17-s + 3.65·19-s − 2.54·20-s + 5.01·22-s + 3.65·23-s − 4.77·25-s − 1.90·26-s + 5.61·28-s − 5.52·29-s + 6.13·31-s + 20.7·32-s − 2.16·34-s − 0.487·35-s − 2.73·37-s + 9.95·38-s − 4.35·40-s + 9.69·41-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 2.70·4-s − 0.210·5-s + 0.392·7-s + 3.28·8-s − 0.404·10-s + 0.555·11-s − 0.194·13-s + 0.755·14-s + 3.61·16-s − 0.192·17-s + 0.839·19-s − 0.568·20-s + 1.06·22-s + 0.761·23-s − 0.955·25-s − 0.374·26-s + 1.06·28-s − 1.02·29-s + 1.10·31-s + 3.66·32-s − 0.370·34-s − 0.0824·35-s − 0.448·37-s + 1.61·38-s − 0.689·40-s + 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4923 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.879032350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.879032350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 547 | \( 1 + T \) |
| good | 2 | \( 1 - 2.72T + 2T^{2} \) |
| 5 | \( 1 + 0.469T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 13 | \( 1 + 0.700T + 13T^{2} \) |
| 17 | \( 1 + 0.793T + 17T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 5.52T + 29T^{2} \) |
| 31 | \( 1 - 6.13T + 31T^{2} \) |
| 37 | \( 1 + 2.73T + 37T^{2} \) |
| 41 | \( 1 - 9.69T + 41T^{2} \) |
| 43 | \( 1 + 4.25T + 43T^{2} \) |
| 47 | \( 1 - 5.21T + 47T^{2} \) |
| 53 | \( 1 - 7.01T + 53T^{2} \) |
| 59 | \( 1 + 6.98T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 + 5.10T + 67T^{2} \) |
| 71 | \( 1 + 1.33T + 71T^{2} \) |
| 73 | \( 1 + 7.83T + 73T^{2} \) |
| 79 | \( 1 - 4.36T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 0.208T + 89T^{2} \) |
| 97 | \( 1 - 3.08T + 97T^{2} \) |
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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
−7.73428366176655917547212491301, −7.42749352293265692776208101979, −6.54600644048941947986903152685, −5.92488581430854232895340617907, −5.19859716209879939831670561449, −4.54932127179405611897662509163, −3.86886024401356812520671148391, −3.14657638178349932582754430249, −2.26961162150532616678188156020, −1.29807842877604857084408390211,
1.29807842877604857084408390211, 2.26961162150532616678188156020, 3.14657638178349932582754430249, 3.86886024401356812520671148391, 4.54932127179405611897662509163, 5.19859716209879939831670561449, 5.92488581430854232895340617907, 6.54600644048941947986903152685, 7.42749352293265692776208101979, 7.73428366176655917547212491301
