| L(s) = 1 | + 2.30·5-s + 7-s + 6.60·13-s − 17-s + 6.60·19-s + 0.302·25-s + 4.30·31-s + 2.30·35-s − 2.60·37-s − 3.90·41-s + 7.30·43-s − 4.60·47-s + 49-s + 3.69·53-s − 9.21·59-s − 7.90·61-s + 15.2·65-s − 1.69·67-s + 7.81·71-s − 7.90·73-s + 12.6·79-s − 6·83-s − 2.30·85-s + 16.6·89-s + 6.60·91-s + 15.2·95-s − 6.30·97-s + ⋯ |
| L(s) = 1 | + 1.02·5-s + 0.377·7-s + 1.83·13-s − 0.242·17-s + 1.51·19-s + 0.0605·25-s + 0.772·31-s + 0.389·35-s − 0.428·37-s − 0.610·41-s + 1.11·43-s − 0.671·47-s + 0.142·49-s + 0.507·53-s − 1.19·59-s − 1.01·61-s + 1.88·65-s − 0.207·67-s + 0.927·71-s − 0.925·73-s + 1.41·79-s − 0.658·83-s − 0.249·85-s + 1.76·89-s + 0.692·91-s + 1.56·95-s − 0.639·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.930751004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.930751004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2.30T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 19 | \( 1 - 6.60T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4.30T + 31T^{2} \) |
| 37 | \( 1 + 2.60T + 37T^{2} \) |
| 41 | \( 1 + 3.90T + 41T^{2} \) |
| 43 | \( 1 - 7.30T + 43T^{2} \) |
| 47 | \( 1 + 4.60T + 47T^{2} \) |
| 53 | \( 1 - 3.69T + 53T^{2} \) |
| 59 | \( 1 + 9.21T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 + 1.69T + 67T^{2} \) |
| 71 | \( 1 - 7.81T + 71T^{2} \) |
| 73 | \( 1 + 7.90T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 + 6.30T + 97T^{2} \) |
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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
−8.413667052895231905269585319245, −7.75151030425490729981516164182, −6.80071084547504318702766169108, −6.07385064255685252344337114756, −5.58713381178498715273151011829, −4.74240679608196964719085534582, −3.74125701247911617777978494011, −2.92800713428309059345534433600, −1.79095117409260482289328532715, −1.07253998566146547319456218513,
1.07253998566146547319456218513, 1.79095117409260482289328532715, 2.92800713428309059345534433600, 3.74125701247911617777978494011, 4.74240679608196964719085534582, 5.58713381178498715273151011829, 6.07385064255685252344337114756, 6.80071084547504318702766169108, 7.75151030425490729981516164182, 8.413667052895231905269585319245
