L(s) = 1 | − 2.21·3-s + 0.305·5-s + 4.25·7-s + 1.90·9-s + 3.66·11-s + 3.96·13-s − 0.677·15-s − 3.21·17-s − 19-s − 9.42·21-s + 2.31·23-s − 4.90·25-s + 2.41·27-s + 7.82·29-s + 10.9·31-s − 8.12·33-s + 1.30·35-s − 2.71·37-s − 8.79·39-s + 4.75·41-s + 0.965·43-s + 0.583·45-s + 7.51·47-s + 11.1·49-s + 7.12·51-s + 5.15·53-s + 1.12·55-s + ⋯ |
L(s) = 1 | − 1.27·3-s + 0.136·5-s + 1.60·7-s + 0.636·9-s + 1.10·11-s + 1.10·13-s − 0.174·15-s − 0.779·17-s − 0.229·19-s − 2.05·21-s + 0.483·23-s − 0.981·25-s + 0.465·27-s + 1.45·29-s + 1.97·31-s − 1.41·33-s + 0.219·35-s − 0.446·37-s − 1.40·39-s + 0.743·41-s + 0.147·43-s + 0.0869·45-s + 1.09·47-s + 1.58·49-s + 0.997·51-s + 0.707·53-s + 0.151·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.925975241\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.925975241\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 + 2.21T + 3T^{2} \) |
| 5 | \( 1 - 0.305T + 5T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 11 | \( 1 - 3.66T + 11T^{2} \) |
| 13 | \( 1 - 3.96T + 13T^{2} \) |
| 17 | \( 1 + 3.21T + 17T^{2} \) |
| 23 | \( 1 - 2.31T + 23T^{2} \) |
| 29 | \( 1 - 7.82T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 2.71T + 37T^{2} \) |
| 41 | \( 1 - 4.75T + 41T^{2} \) |
| 43 | \( 1 - 0.965T + 43T^{2} \) |
| 47 | \( 1 - 7.51T + 47T^{2} \) |
| 53 | \( 1 - 5.15T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 7.54T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + 5.21T + 73T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 - 1.30T + 89T^{2} \) |
| 97 | \( 1 + 0.304T + 97T^{2} \) |
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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
−8.283066127288036578949607719324, −7.19377805037393026146482081240, −6.45325163498866597548313448891, −6.01772630997774233934670940216, −5.23677220619877329705036807640, −4.45787474980019656246779720822, −4.10637132979562891022570996010, −2.63765592356891291805275167302, −1.47709472560030345689981862200, −0.892258651730687450606925904306,
0.892258651730687450606925904306, 1.47709472560030345689981862200, 2.63765592356891291805275167302, 4.10637132979562891022570996010, 4.45787474980019656246779720822, 5.23677220619877329705036807640, 6.01772630997774233934670940216, 6.45325163498866597548313448891, 7.19377805037393026146482081240, 8.283066127288036578949607719324