| L(s) = 1 | − 3-s + 3.90·5-s − 0.0794·7-s + 9-s + 1.50·11-s − 3.90·15-s − 6.52·17-s − 0.786·19-s + 0.0794·21-s − 7.03·23-s + 10.2·25-s − 27-s + 6.15·29-s + 1.43·31-s − 1.50·33-s − 0.310·35-s + 9.23·37-s + 7.75·41-s + 1.06·43-s + 3.90·45-s + 12.7·47-s − 6.99·49-s + 6.52·51-s + 2.73·53-s + 5.87·55-s + 0.786·57-s + 10.3·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.74·5-s − 0.0300·7-s + 0.333·9-s + 0.453·11-s − 1.00·15-s − 1.58·17-s − 0.180·19-s + 0.0173·21-s − 1.46·23-s + 2.05·25-s − 0.192·27-s + 1.14·29-s + 0.258·31-s − 0.261·33-s − 0.0524·35-s + 1.51·37-s + 1.21·41-s + 0.162·43-s + 0.582·45-s + 1.86·47-s − 0.999·49-s + 0.913·51-s + 0.375·53-s + 0.792·55-s + 0.104·57-s + 1.35·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.224294472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.224294472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 + 0.0794T + 7T^{2} \) |
| 11 | \( 1 - 1.50T + 11T^{2} \) |
| 17 | \( 1 + 6.52T + 17T^{2} \) |
| 19 | \( 1 + 0.786T + 19T^{2} \) |
| 23 | \( 1 + 7.03T + 23T^{2} \) |
| 29 | \( 1 - 6.15T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 - 7.75T + 41T^{2} \) |
| 43 | \( 1 - 1.06T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 2.73T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 + 7.80T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 4.16T + 73T^{2} \) |
| 79 | \( 1 - 3.83T + 79T^{2} \) |
| 83 | \( 1 - 9.04T + 83T^{2} \) |
| 89 | \( 1 + 4.75T + 89T^{2} \) |
| 97 | \( 1 - 6.94T + 97T^{2} \) |
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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
−8.707027002042303024339256351919, −7.59478879873140696183450258856, −6.59981500390288502395728717651, −6.21912437987065829024662904819, −5.70510412480194571157169445816, −4.71382343312529751392212527602, −4.10101062517293584525930759548, −2.57058705737499975059123353526, −2.05045789582700608954420073242, −0.895509657725381405572484338726,
0.895509657725381405572484338726, 2.05045789582700608954420073242, 2.57058705737499975059123353526, 4.10101062517293584525930759548, 4.71382343312529751392212527602, 5.70510412480194571157169445816, 6.21912437987065829024662904819, 6.59981500390288502395728717651, 7.59478879873140696183450258856, 8.707027002042303024339256351919
