| L(s) = 1 | + 1.31·2-s + 3-s − 0.282·4-s − 0.382·5-s + 1.31·6-s − 2.51·7-s − 2.99·8-s + 9-s − 0.501·10-s − 2.09·11-s − 0.282·12-s + 3.30·13-s − 3.29·14-s − 0.382·15-s − 3.35·16-s + 17-s + 1.31·18-s + 4.13·19-s + 0.108·20-s − 2.51·21-s − 2.74·22-s − 3.29·23-s − 2.99·24-s − 4.85·25-s + 4.33·26-s + 27-s + 0.710·28-s + ⋯ |
| L(s) = 1 | + 0.926·2-s + 0.577·3-s − 0.141·4-s − 0.171·5-s + 0.535·6-s − 0.951·7-s − 1.05·8-s + 0.333·9-s − 0.158·10-s − 0.632·11-s − 0.0814·12-s + 0.917·13-s − 0.881·14-s − 0.0988·15-s − 0.838·16-s + 0.242·17-s + 0.308·18-s + 0.948·19-s + 0.0241·20-s − 0.549·21-s − 0.586·22-s − 0.686·23-s − 0.610·24-s − 0.970·25-s + 0.850·26-s + 0.192·27-s + 0.134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.648542123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.648542123\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 1.31T + 2T^{2} \) |
| 5 | \( 1 + 0.382T + 5T^{2} \) |
| 7 | \( 1 + 2.51T + 7T^{2} \) |
| 11 | \( 1 + 2.09T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 19 | \( 1 - 4.13T + 19T^{2} \) |
| 23 | \( 1 + 3.29T + 23T^{2} \) |
| 29 | \( 1 - 6.91T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 + 3.87T + 41T^{2} \) |
| 43 | \( 1 + 7.05T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 6.71T + 59T^{2} \) |
| 61 | \( 1 + 3.61T + 61T^{2} \) |
| 67 | \( 1 + 4.26T + 67T^{2} \) |
| 71 | \( 1 + 2.45T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 2.07T + 89T^{2} \) |
| 97 | \( 1 + 9.19T + 97T^{2} \) |
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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
−8.378720172308934954497214131088, −7.84793791909780192380845185284, −6.73908796877940433058608057912, −6.15806636459102890717814895315, −5.42158005464490154966169106193, −4.53826288536424092628990862465, −3.75152927607198614008900900490, −3.19093794249563729435881547905, −2.44177047413251887166976859934, −0.78008025842949259207948904758,
0.78008025842949259207948904758, 2.44177047413251887166976859934, 3.19093794249563729435881547905, 3.75152927607198614008900900490, 4.53826288536424092628990862465, 5.42158005464490154966169106193, 6.15806636459102890717814895315, 6.73908796877940433058608057912, 7.84793791909780192380845185284, 8.378720172308934954497214131088
