| L(s) = 1 | + 2-s + 1.82·3-s + 4-s − 5-s + 1.82·6-s − 3.76·7-s + 8-s + 0.326·9-s − 10-s − 11-s + 1.82·12-s + 6.64·13-s − 3.76·14-s − 1.82·15-s + 16-s + 1.83·17-s + 0.326·18-s − 2.41·19-s − 20-s − 6.86·21-s − 22-s − 7.65·23-s + 1.82·24-s + 25-s + 6.64·26-s − 4.87·27-s − 3.76·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.05·3-s + 0.5·4-s − 0.447·5-s + 0.744·6-s − 1.42·7-s + 0.353·8-s + 0.108·9-s − 0.316·10-s − 0.301·11-s + 0.526·12-s + 1.84·13-s − 1.00·14-s − 0.470·15-s + 0.250·16-s + 0.443·17-s + 0.0768·18-s − 0.554·19-s − 0.223·20-s − 1.49·21-s − 0.213·22-s − 1.59·23-s + 0.372·24-s + 0.200·25-s + 1.30·26-s − 0.938·27-s − 0.711·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 1.82T + 3T^{2} \) |
| 7 | \( 1 + 3.76T + 7T^{2} \) |
| 13 | \( 1 - 6.64T + 13T^{2} \) |
| 17 | \( 1 - 1.83T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 5.16T + 29T^{2} \) |
| 31 | \( 1 + 8.41T + 31T^{2} \) |
| 37 | \( 1 + 6.78T + 37T^{2} \) |
| 41 | \( 1 - 0.709T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 9.81T + 61T^{2} \) |
| 67 | \( 1 - 5.05T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 79 | \( 1 + 6.93T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 7.69T + 97T^{2} \) |
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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
−7.49358427361849781566467153271, −6.72375530007860037723883925694, −6.01638665726890835338551936238, −5.63289972441403348461824616780, −4.25919737100134200493100736505, −3.71932214058847454724689594897, −3.29205366347906707566147529625, −2.58508781189546131507889490581, −1.55750655576093968776403867535, 0,
1.55750655576093968776403867535, 2.58508781189546131507889490581, 3.29205366347906707566147529625, 3.71932214058847454724689594897, 4.25919737100134200493100736505, 5.63289972441403348461824616780, 6.01638665726890835338551936238, 6.72375530007860037723883925694, 7.49358427361849781566467153271
