| L(s) = 1 | − 5.15·2-s + 18.5·4-s − 7·7-s − 54.1·8-s − 14.0·11-s + 10.5·13-s + 36.0·14-s + 130.·16-s − 131.·17-s + 38.6·19-s + 72.1·22-s + 102.·23-s − 54.5·26-s − 129.·28-s − 232.·29-s − 165.·31-s − 240.·32-s + 678.·34-s − 280.·37-s − 199.·38-s − 122.·41-s + 431.·43-s − 259.·44-s − 527.·46-s − 295.·47-s + 49·49-s + 196.·52-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 2.31·4-s − 0.377·7-s − 2.39·8-s − 0.383·11-s + 0.225·13-s + 0.688·14-s + 2.04·16-s − 1.88·17-s + 0.467·19-s + 0.699·22-s + 0.929·23-s − 0.411·26-s − 0.875·28-s − 1.48·29-s − 0.958·31-s − 1.32·32-s + 3.42·34-s − 1.24·37-s − 0.850·38-s − 0.465·41-s + 1.53·43-s − 0.888·44-s − 1.69·46-s − 0.917·47-s + 0.142·49-s + 0.523·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4105799737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4105799737\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 5.15T + 8T^{2} \) |
| 11 | \( 1 + 14.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 232.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 280.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 122.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 431.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 295.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 243.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 566.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 188.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 871.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 176.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 220.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 190.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 518.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 598.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.88e3T + 9.12e5T^{2} \) |
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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
−9.035054463349301451873375627657, −8.548631268179720979739734941663, −7.50898336244962145328208086782, −7.00578654911884829737322937340, −6.22782504886407723252194636500, −5.13793232213379700265987231704, −3.67131156180931450879244246111, −2.52730574227726522213851509134, −1.68581771018333395920913492289, −0.39913220666423494781514273490,
0.39913220666423494781514273490, 1.68581771018333395920913492289, 2.52730574227726522213851509134, 3.67131156180931450879244246111, 5.13793232213379700265987231704, 6.22782504886407723252194636500, 7.00578654911884829737322937340, 7.50898336244962145328208086782, 8.548631268179720979739734941663, 9.035054463349301451873375627657
