L(s) = 1 | − 166.·2-s − 5.90e4·3-s − 2.06e6·4-s + 9.84e6·6-s − 1.02e9·7-s + 6.94e8·8-s + 3.48e9·9-s + 8.41e10·11-s + 1.22e11·12-s − 6.78e11·13-s + 1.70e11·14-s + 4.22e12·16-s + 5.16e12·17-s − 5.81e11·18-s + 5.03e12·19-s + 6.05e13·21-s − 1.40e13·22-s − 3.20e14·23-s − 4.10e13·24-s + 1.13e14·26-s − 2.05e14·27-s + 2.12e15·28-s + 3.64e15·29-s − 4.68e15·31-s − 2.16e15·32-s − 4.96e15·33-s − 8.61e14·34-s + ⋯ |
L(s) = 1 | − 0.115·2-s − 0.577·3-s − 0.986·4-s + 0.0664·6-s − 1.37·7-s + 0.228·8-s + 0.333·9-s + 0.978·11-s + 0.569·12-s − 1.36·13-s + 0.157·14-s + 0.960·16-s + 0.621·17-s − 0.0383·18-s + 0.188·19-s + 0.791·21-s − 0.112·22-s − 1.61·23-s − 0.132·24-s + 0.157·26-s − 0.192·27-s + 1.35·28-s + 1.61·29-s − 1.02·31-s − 0.339·32-s − 0.564·33-s − 0.0715·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(0.1518092341\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1518092341\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.90e4T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 166.T + 2.09e6T^{2} \) |
| 7 | \( 1 + 1.02e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 8.41e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 6.78e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 5.16e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 5.03e12T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.20e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 3.64e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 4.68e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 5.20e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 7.94e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.06e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 1.15e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 2.45e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 7.28e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.31e17T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.83e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.72e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 3.44e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 2.81e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.06e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 8.54e19T + 8.65e40T^{2} \) |
| 97 | \( 1 + 7.33e20T + 5.27e41T^{2} \) |
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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
−10.14352927514448129738187320394, −9.828450946689990016007379849914, −8.721101200261689507887745780328, −7.30783269493095119381885092596, −6.28272600103872524992339409151, −5.22015136774598228374322197954, −4.11487468476220755613911483374, −3.15904255633060398457059751210, −1.49525516333199497988170751836, −0.17631403898533137505773880277,
0.17631403898533137505773880277, 1.49525516333199497988170751836, 3.15904255633060398457059751210, 4.11487468476220755613911483374, 5.22015136774598228374322197954, 6.28272600103872524992339409151, 7.30783269493095119381885092596, 8.721101200261689507887745780328, 9.828450946689990016007379849914, 10.14352927514448129738187320394