| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 11-s − 2·13-s + 16-s − 20-s + 22-s − 2·23-s + 25-s + 2·26-s + 2·31-s − 32-s − 6·37-s + 40-s + 6·41-s + 6·43-s − 44-s + 2·46-s + 6·47-s − 7·49-s − 50-s − 2·52-s − 6·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s − 0.223·20-s + 0.213·22-s − 0.417·23-s + 1/5·25-s + 0.392·26-s + 0.359·31-s − 0.176·32-s − 0.986·37-s + 0.158·40-s + 0.937·41-s + 0.914·43-s − 0.150·44-s + 0.294·46-s + 0.875·47-s − 49-s − 0.141·50-s − 0.277·52-s − 0.824·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.81264661787445, −12.35305601398423, −12.14128132669308, −11.53834639058608, −11.16529067651015, −10.54857016827209, −10.40243293426278, −9.786340333889490, −9.281469861951604, −8.942878436256693, −8.446337790014393, −7.792323521016109, −7.641992272178051, −7.186368947226870, −6.548297496370772, −6.121909840663331, −5.590446278587806, −5.032922459117325, −4.411300925887008, −4.068830353001582, −3.258034142959954, −2.831754480834087, −2.347313684132923, −1.563772588608883, −1.122283146136363, 0, 0,
1.122283146136363, 1.563772588608883, 2.347313684132923, 2.831754480834087, 3.258034142959954, 4.068830353001582, 4.411300925887008, 5.032922459117325, 5.590446278587806, 6.121909840663331, 6.548297496370772, 7.186368947226870, 7.641992272178051, 7.792323521016109, 8.446337790014393, 8.942878436256693, 9.281469861951604, 9.786340333889490, 10.40243293426278, 10.54857016827209, 11.16529067651015, 11.53834639058608, 12.14128132669308, 12.35305601398423, 12.81264661787445
