| L(s) = 1 | − 2·4-s + 4·5-s − 7-s − 11-s + 13-s + 4·16-s + 17-s − 8·20-s − 8·23-s + 11·25-s + 2·28-s + 2·29-s + 2·31-s − 4·35-s + 4·37-s + 6·43-s + 2·44-s − 4·47-s + 49-s − 2·52-s + 53-s − 4·55-s − 7·59-s + 8·61-s − 8·64-s + 4·65-s − 4·67-s + ⋯ |
| L(s) = 1 | − 4-s + 1.78·5-s − 0.377·7-s − 0.301·11-s + 0.277·13-s + 16-s + 0.242·17-s − 1.78·20-s − 1.66·23-s + 11/5·25-s + 0.377·28-s + 0.371·29-s + 0.359·31-s − 0.676·35-s + 0.657·37-s + 0.914·43-s + 0.301·44-s − 0.583·47-s + 1/7·49-s − 0.277·52-s + 0.137·53-s − 0.539·55-s − 0.911·59-s + 1.02·61-s − 64-s + 0.496·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.16568536319385, −12.87387054044345, −12.17403581747068, −11.97778589206220, −11.04103120589024, −10.57935243577037, −10.08802005478686, −9.885499037542848, −9.463222675024362, −8.952312782963393, −8.615232295945469, −7.916759599867256, −7.620549794313027, −6.712657422605944, −6.248424259540905, −5.946507914227123, −5.457018424415554, −5.036958723716091, −4.376954879218345, −3.946272251901111, −3.182218567118155, −2.674742899441071, −2.084292266571741, −1.445700071833856, −0.8593287449263464, 0,
0.8593287449263464, 1.445700071833856, 2.084292266571741, 2.674742899441071, 3.182218567118155, 3.946272251901111, 4.376954879218345, 5.036958723716091, 5.457018424415554, 5.946507914227123, 6.248424259540905, 6.712657422605944, 7.620549794313027, 7.916759599867256, 8.615232295945469, 8.952312782963393, 9.463222675024362, 9.885499037542848, 10.08802005478686, 10.57935243577037, 11.04103120589024, 11.97778589206220, 12.17403581747068, 12.87387054044345, 13.16568536319385
