| L(s) = 1 | − 7.33·3-s + 214.·5-s − 571.·7-s − 675.·9-s − 2.19e3·13-s − 1.57e3·15-s − 9.63e3·17-s + 4.19e3·21-s + 3.05e4·25-s + 1.03e4·27-s + 2.78e4·31-s − 1.22e5·35-s + 7.99e4·37-s + 1.61e4·39-s + 4.24e4·43-s − 1.45e5·45-s + 2.05e5·47-s + 2.09e5·49-s + 7.07e4·51-s + 3.86e5·63-s − 4.72e5·65-s − 7.14e5·71-s − 2.24e5·75-s + 4.16e5·81-s − 2.07e6·85-s + 1.25e6·91-s − 2.04e5·93-s + ⋯ |
| L(s) = 1 | − 0.271·3-s + 1.71·5-s − 1.66·7-s − 0.926·9-s − 13-s − 0.467·15-s − 1.96·17-s + 0.453·21-s + 1.95·25-s + 0.523·27-s + 0.934·31-s − 2.86·35-s + 1.57·37-s + 0.271·39-s + 0.533·43-s − 1.59·45-s + 1.97·47-s + 1.78·49-s + 0.533·51-s + 1.54·63-s − 1.71·65-s − 1.99·71-s − 0.531·75-s + 0.783·81-s − 3.37·85-s + 1.66·91-s − 0.253·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.388130455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.388130455\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 3 | \( 1 + 7.33T + 729T^{2} \) |
| 5 | \( 1 - 214.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 571.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.77e6T^{2} \) |
| 17 | \( 1 + 9.63e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.48e8T^{2} \) |
| 29 | \( 1 - 5.94e8T^{2} \) |
| 31 | \( 1 - 2.78e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 7.99e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 4.75e9T^{2} \) |
| 43 | \( 1 - 4.24e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 2.05e5T + 1.07e10T^{2} \) |
| 53 | \( 1 - 2.21e10T^{2} \) |
| 59 | \( 1 - 4.21e10T^{2} \) |
| 61 | \( 1 - 5.15e10T^{2} \) |
| 67 | \( 1 - 9.04e10T^{2} \) |
| 71 | \( 1 + 7.14e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.43e11T^{2} \) |
| 83 | \( 1 - 3.26e11T^{2} \) |
| 89 | \( 1 - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.32e11T^{2} \) |
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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
−10.07992138620381170639806027579, −9.377494799437826969123263268867, −8.819526307071400331354765961406, −7.06160332394129968315041982068, −6.23686324897468646551939786324, −5.80498711493180292741602862185, −4.53841345467309530569612245774, −2.79555366710034490608897012609, −2.33681847047763002359006457511, −0.54402052549489841218121144042,
0.54402052549489841218121144042, 2.33681847047763002359006457511, 2.79555366710034490608897012609, 4.53841345467309530569612245774, 5.80498711493180292741602862185, 6.23686324897468646551939786324, 7.06160332394129968315041982068, 8.819526307071400331354765961406, 9.377494799437826969123263268867, 10.07992138620381170639806027579
