| L(s) = 1 | + 346.·2-s − 2.18e3·3-s + 8.70e4·4-s − 7.57e5·6-s − 4.02e6·7-s + 1.88e7·8-s + 4.78e6·9-s − 2.10e7·11-s − 1.90e8·12-s − 1.69e6·13-s − 1.39e9·14-s + 3.65e9·16-s + 3.24e9·17-s + 1.65e9·18-s − 1.81e9·19-s + 8.81e9·21-s − 7.28e9·22-s + 1.20e10·23-s − 4.11e10·24-s − 5.86e8·26-s − 1.04e10·27-s − 3.50e11·28-s + 1.00e11·29-s + 1.92e11·31-s + 6.49e11·32-s + 4.60e10·33-s + 1.12e12·34-s + ⋯ |
| L(s) = 1 | + 1.91·2-s − 0.577·3-s + 2.65·4-s − 1.10·6-s − 1.84·7-s + 3.17·8-s + 0.333·9-s − 0.325·11-s − 1.53·12-s − 0.00748·13-s − 3.53·14-s + 3.40·16-s + 1.91·17-s + 0.637·18-s − 0.465·19-s + 1.06·21-s − 0.622·22-s + 0.735·23-s − 1.83·24-s − 0.0143·26-s − 0.192·27-s − 4.91·28-s + 1.08·29-s + 1.25·31-s + 3.34·32-s + 0.188·33-s + 3.66·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(5.978634451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.978634451\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 2.18e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 346.T + 3.27e4T^{2} \) |
| 7 | \( 1 + 4.02e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 2.10e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 1.69e6T + 5.11e16T^{2} \) |
| 17 | \( 1 - 3.24e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 1.81e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 1.20e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 1.00e11T + 8.62e21T^{2} \) |
| 31 | \( 1 - 1.92e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 1.21e10T + 3.33e23T^{2} \) |
| 41 | \( 1 - 1.74e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.51e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 3.60e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 4.83e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 1.68e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 2.26e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 2.44e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 9.17e12T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.13e14T + 8.90e27T^{2} \) |
| 79 | \( 1 - 2.09e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 1.53e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 3.76e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 6.17e14T + 6.33e29T^{2} \) |
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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
−12.14589712461645786890896371781, −10.74874821718972767323208122855, −9.844803831729837568917140049289, −7.49965548819385332144238224187, −6.39328947067435460738543414550, −5.84702837220241107474467015676, −4.63595537942585791984938957562, −3.40352601526549939921639259659, −2.72754118563661273111741728472, −0.907588625232290170827438582551,
0.907588625232290170827438582551, 2.72754118563661273111741728472, 3.40352601526549939921639259659, 4.63595537942585791984938957562, 5.84702837220241107474467015676, 6.39328947067435460738543414550, 7.49965548819385332144238224187, 9.844803831729837568917140049289, 10.74874821718972767323208122855, 12.14589712461645786890896371781
