| L(s) = 1 | − 5-s + 5.21·7-s − 2.67·11-s + 0.186·13-s + 1.99·17-s − 0.675·19-s − 23-s + 25-s − 5.02·29-s + 7.52·31-s − 5.21·35-s + 4.34·37-s + 6.60·41-s − 2.84·43-s + 13.5·47-s + 20.1·49-s − 12.9·53-s + 2.67·55-s + 6.74·59-s + 8.65·61-s − 0.186·65-s − 12.1·67-s − 0.181·71-s + 3.81·73-s − 13.9·77-s − 10.4·79-s − 16.8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.96·7-s − 0.806·11-s + 0.0516·13-s + 0.482·17-s − 0.154·19-s − 0.208·23-s + 0.200·25-s − 0.933·29-s + 1.35·31-s − 0.880·35-s + 0.715·37-s + 1.03·41-s − 0.433·43-s + 1.98·47-s + 2.87·49-s − 1.77·53-s + 0.360·55-s + 0.878·59-s + 1.10·61-s − 0.0230·65-s − 1.48·67-s − 0.0215·71-s + 0.446·73-s − 1.58·77-s − 1.18·79-s − 1.84·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.207822952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.207822952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 5.21T + 7T^{2} \) |
| 11 | \( 1 + 2.67T + 11T^{2} \) |
| 13 | \( 1 - 0.186T + 13T^{2} \) |
| 17 | \( 1 - 1.99T + 17T^{2} \) |
| 19 | \( 1 + 0.675T + 19T^{2} \) |
| 29 | \( 1 + 5.02T + 29T^{2} \) |
| 31 | \( 1 - 7.52T + 31T^{2} \) |
| 37 | \( 1 - 4.34T + 37T^{2} \) |
| 41 | \( 1 - 6.60T + 41T^{2} \) |
| 43 | \( 1 + 2.84T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 6.74T + 59T^{2} \) |
| 61 | \( 1 - 8.65T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 0.181T + 71T^{2} \) |
| 73 | \( 1 - 3.81T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−8.309791505957774227457205620427, −7.67768171626550087172233117436, −7.35361636175555433797887925314, −6.04352894037190128275474013796, −5.36205182254076985060754456953, −4.62430979141554847053993184274, −4.07198848856496717243885142209, −2.82434371512339027520239965480, −1.93845225476007227289436983356, −0.879566558021662312461146091902,
0.879566558021662312461146091902, 1.93845225476007227289436983356, 2.82434371512339027520239965480, 4.07198848856496717243885142209, 4.62430979141554847053993184274, 5.36205182254076985060754456953, 6.04352894037190128275474013796, 7.35361636175555433797887925314, 7.67768171626550087172233117436, 8.309791505957774227457205620427
