L(s) = 1 | − 1.43·3-s − 5-s + 7-s − 0.950·9-s − 13-s + 1.43·15-s − 3.95·17-s − 5.95·19-s − 1.43·21-s − 5.03·23-s + 25-s + 5.65·27-s + 4.46·29-s + 2.56·31-s − 35-s + 5.60·37-s + 1.43·39-s − 3.95·41-s − 1.13·43-s + 0.950·45-s + 9.03·47-s + 49-s + 5.65·51-s + 2·53-s + 8.51·57-s − 0.294·59-s − 10·61-s + ⋯ |
L(s) = 1 | − 0.826·3-s − 0.447·5-s + 0.377·7-s − 0.316·9-s − 0.277·13-s + 0.369·15-s − 0.958·17-s − 1.36·19-s − 0.312·21-s − 1.05·23-s + 0.200·25-s + 1.08·27-s + 0.829·29-s + 0.461·31-s − 0.169·35-s + 0.921·37-s + 0.229·39-s − 0.616·41-s − 0.173·43-s + 0.141·45-s + 1.31·47-s + 0.142·49-s + 0.791·51-s + 0.274·53-s + 1.12·57-s − 0.0383·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7612575387\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7612575387\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 1.43T + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 3.95T + 17T^{2} \) |
| 19 | \( 1 + 5.95T + 19T^{2} \) |
| 23 | \( 1 + 5.03T + 23T^{2} \) |
| 29 | \( 1 - 4.46T + 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 + 3.95T + 41T^{2} \) |
| 43 | \( 1 + 1.13T + 43T^{2} \) |
| 47 | \( 1 - 9.03T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 0.294T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 8.12T + 67T^{2} \) |
| 71 | \( 1 - 2.86T + 71T^{2} \) |
| 73 | \( 1 + 8.76T + 73T^{2} \) |
| 79 | \( 1 - 5.95T + 79T^{2} \) |
| 83 | \( 1 - 5.03T + 83T^{2} \) |
| 89 | \( 1 - 2.74T + 89T^{2} \) |
| 97 | \( 1 + 9.21T + 97T^{2} \) |
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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
−8.464108276767221287870821724381, −7.898847017667118859778103230071, −6.89820170844058866871540719525, −6.28566543851752052196195069280, −5.63156341558429639577363175206, −4.57076375070863188575331311390, −4.27971420414855641776744818235, −2.93150573880115084134502196142, −1.98858395828053413216408193115, −0.51233293304449176811288634343,
0.51233293304449176811288634343, 1.98858395828053413216408193115, 2.93150573880115084134502196142, 4.27971420414855641776744818235, 4.57076375070863188575331311390, 5.63156341558429639577363175206, 6.28566543851752052196195069280, 6.89820170844058866871540719525, 7.898847017667118859778103230071, 8.464108276767221287870821724381