L(s) = 1 | − 3-s + 7-s + 9-s + 4·11-s − 13-s − 7·17-s + 2·19-s − 21-s + 5·23-s − 27-s + 4·29-s − 4·33-s + 39-s − 12·41-s + 4·43-s − 9·47-s + 49-s + 7·51-s − 10·53-s − 2·57-s + 3·59-s − 4·61-s + 63-s − 3·67-s − 5·69-s − 16·71-s − 5·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 1.69·17-s + 0.458·19-s − 0.218·21-s + 1.04·23-s − 0.192·27-s + 0.742·29-s − 0.696·33-s + 0.160·39-s − 1.87·41-s + 0.609·43-s − 1.31·47-s + 1/7·49-s + 0.980·51-s − 1.37·53-s − 0.264·57-s + 0.390·59-s − 0.512·61-s + 0.125·63-s − 0.366·67-s − 0.601·69-s − 1.89·71-s − 0.585·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.358658244\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.358658244\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.74985727530044, −13.03474634000654, −12.78543188231799, −12.04609435835516, −11.63276923608151, −11.34442164700052, −10.88404847401928, −10.23665507241587, −9.821504739224465, −9.085284460311801, −8.866777208374038, −8.306301436612451, −7.561889521201831, −7.020676058966201, −6.585281050469200, −6.276230301593583, −5.481508557552774, −4.880360587830950, −4.531476767043735, −4.006273591550883, −3.193863081478628, −2.662468212488339, −1.619003028002422, −1.437672166637552, −0.3767233034319698,
0.3767233034319698, 1.437672166637552, 1.619003028002422, 2.662468212488339, 3.193863081478628, 4.006273591550883, 4.531476767043735, 4.880360587830950, 5.481508557552774, 6.276230301593583, 6.585281050469200, 7.020676058966201, 7.561889521201831, 8.306301436612451, 8.866777208374038, 9.085284460311801, 9.821504739224465, 10.23665507241587, 10.88404847401928, 11.34442164700052, 11.63276923608151, 12.04609435835516, 12.78543188231799, 13.03474634000654, 13.74985727530044