| L(s) = 1 | + 0.475·2-s + 3-s − 1.77·4-s + 5-s + 0.475·6-s − 1.79·8-s + 9-s + 0.475·10-s − 11-s − 1.77·12-s + 6.31·13-s + 15-s + 2.69·16-s − 1.92·17-s + 0.475·18-s + 3.92·19-s − 1.77·20-s − 0.475·22-s + 4.74·23-s − 1.79·24-s + 25-s + 3.00·26-s + 27-s − 4.08·29-s + 0.475·30-s − 7.20·31-s + 4.87·32-s + ⋯ |
| L(s) = 1 | + 0.336·2-s + 0.577·3-s − 0.886·4-s + 0.447·5-s + 0.194·6-s − 0.634·8-s + 0.333·9-s + 0.150·10-s − 0.301·11-s − 0.511·12-s + 1.75·13-s + 0.258·15-s + 0.673·16-s − 0.466·17-s + 0.112·18-s + 0.899·19-s − 0.396·20-s − 0.101·22-s + 0.988·23-s − 0.366·24-s + 0.200·25-s + 0.589·26-s + 0.192·27-s − 0.757·29-s + 0.0868·30-s − 1.29·31-s + 0.861·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.877639784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.877639784\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.475T + 2T^{2} \) |
| 13 | \( 1 - 6.31T + 13T^{2} \) |
| 17 | \( 1 + 1.92T + 17T^{2} \) |
| 19 | \( 1 - 3.92T + 19T^{2} \) |
| 23 | \( 1 - 4.74T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 + 7.20T + 31T^{2} \) |
| 37 | \( 1 - 9.04T + 37T^{2} \) |
| 41 | \( 1 + 2.67T + 41T^{2} \) |
| 43 | \( 1 - 3.43T + 43T^{2} \) |
| 47 | \( 1 + 9.44T + 47T^{2} \) |
| 53 | \( 1 + 4.77T + 53T^{2} \) |
| 59 | \( 1 - 7.04T + 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 + 6.01T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 + 1.69T + 79T^{2} \) |
| 83 | \( 1 + 4.23T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.007355056351042505023428785747, −7.15671808302397879480723261627, −6.31985802647969357554866493176, −5.63031720207095306141711488071, −5.06727579078642285645054979901, −4.16531320772020106907039686879, −3.53708095530301341227869819615, −2.92452953173670369837921352710, −1.75371611594454598261484016467, −0.816139885827433341431602264104,
0.816139885827433341431602264104, 1.75371611594454598261484016467, 2.92452953173670369837921352710, 3.53708095530301341227869819615, 4.16531320772020106907039686879, 5.06727579078642285645054979901, 5.63031720207095306141711488071, 6.31985802647969357554866493176, 7.15671808302397879480723261627, 8.007355056351042505023428785747
