| L(s) = 1 | + 2-s − 2.18·3-s + 4-s + 4.29·5-s − 2.18·6-s + 2.69·7-s + 8-s + 1.77·9-s + 4.29·10-s + 3.71·11-s − 2.18·12-s + 2.69·14-s − 9.37·15-s + 16-s − 0.184·17-s + 1.77·18-s − 19-s + 4.29·20-s − 5.88·21-s + 3.71·22-s − 6.17·23-s − 2.18·24-s + 13.4·25-s + 2.68·27-s + 2.69·28-s − 1.34·29-s − 9.37·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.26·3-s + 0.5·4-s + 1.91·5-s − 0.891·6-s + 1.01·7-s + 0.353·8-s + 0.591·9-s + 1.35·10-s + 1.12·11-s − 0.630·12-s + 0.720·14-s − 2.42·15-s + 0.250·16-s − 0.0448·17-s + 0.417·18-s − 0.229·19-s + 0.959·20-s − 1.28·21-s + 0.792·22-s − 1.28·23-s − 0.445·24-s + 2.68·25-s + 0.515·27-s + 0.509·28-s − 0.250·29-s − 1.71·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.846135982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.846135982\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.18T + 3T^{2} \) |
| 5 | \( 1 - 4.29T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 17 | \( 1 + 0.184T + 17T^{2} \) |
| 23 | \( 1 + 6.17T + 23T^{2} \) |
| 29 | \( 1 + 1.34T + 29T^{2} \) |
| 31 | \( 1 - 1.92T + 31T^{2} \) |
| 37 | \( 1 - 7.75T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 6.82T + 43T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 + 5.59T + 53T^{2} \) |
| 59 | \( 1 - 2.45T + 59T^{2} \) |
| 61 | \( 1 - 7.30T + 61T^{2} \) |
| 67 | \( 1 - 15.0T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 + 4.04T + 73T^{2} \) |
| 79 | \( 1 - 6.12T + 79T^{2} \) |
| 83 | \( 1 - 7.49T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 7.23T + 97T^{2} \) |
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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
−7.941100546120580557006379471345, −6.68258728204231522143753058574, −6.49855061949557136721218450185, −5.79349760185345693707188690376, −5.30603143777245763012723860679, −4.72553714668272294439990694571, −3.89545023952815320703422746031, −2.52444355371668463734968455362, −1.79006374304906427783909827884, −1.06975424354930776113209611247,
1.06975424354930776113209611247, 1.79006374304906427783909827884, 2.52444355371668463734968455362, 3.89545023952815320703422746031, 4.72553714668272294439990694571, 5.30603143777245763012723860679, 5.79349760185345693707188690376, 6.49855061949557136721218450185, 6.68258728204231522143753058574, 7.941100546120580557006379471345
