| L(s) = 1 | − 5.40·3-s + 9.23·5-s − 2.17·7-s + 2.26·9-s + 51.8·11-s − 10.4·13-s − 49.9·15-s + 17·17-s + 122.·19-s + 11.7·21-s − 48.3·23-s − 39.6·25-s + 133.·27-s − 160.·29-s − 12.2·31-s − 280.·33-s − 20.0·35-s + 103.·37-s + 56.3·39-s + 113.·41-s + 76.1·43-s + 20.8·45-s + 289.·47-s − 338.·49-s − 91.9·51-s − 447.·53-s + 479.·55-s + ⋯ |
| L(s) = 1 | − 1.04·3-s + 0.826·5-s − 0.117·7-s + 0.0837·9-s + 1.42·11-s − 0.222·13-s − 0.860·15-s + 0.242·17-s + 1.47·19-s + 0.121·21-s − 0.438·23-s − 0.317·25-s + 0.953·27-s − 1.02·29-s − 0.0708·31-s − 1.48·33-s − 0.0968·35-s + 0.458·37-s + 0.231·39-s + 0.431·41-s + 0.270·43-s + 0.0692·45-s + 0.899·47-s − 0.986·49-s − 0.252·51-s − 1.15·53-s + 1.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.710181740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.710181740\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 + 5.40T + 27T^{2} \) |
| 5 | \( 1 - 9.23T + 125T^{2} \) |
| 7 | \( 1 + 2.17T + 343T^{2} \) |
| 11 | \( 1 - 51.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10.4T + 2.19e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 48.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 12.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 103.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 76.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 289.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 447.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 480.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 56.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 332.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 35.2T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 471.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 559.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 869.T + 9.12e5T^{2} \) |
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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
−9.598164972671707101592130228111, −8.934139291139403617239710809432, −7.65498562994278175669243484205, −6.76195766429063034288246823498, −5.92346349569100770697181741457, −5.52170112609703579109032179844, −4.38694856053917830908231077548, −3.23681171997058057222987907533, −1.80472432553858254570618460014, −0.74687511216339550064586603789,
0.74687511216339550064586603789, 1.80472432553858254570618460014, 3.23681171997058057222987907533, 4.38694856053917830908231077548, 5.52170112609703579109032179844, 5.92346349569100770697181741457, 6.76195766429063034288246823498, 7.65498562994278175669243484205, 8.934139291139403617239710809432, 9.598164972671707101592130228111
