| L(s) = 1 | + 2.47·2-s − 3-s + 4.13·4-s + 3.02·5-s − 2.47·6-s − 7-s + 5.27·8-s + 9-s + 7.49·10-s − 2.35·11-s − 4.13·12-s + 2.34·13-s − 2.47·14-s − 3.02·15-s + 4.80·16-s − 2.25·17-s + 2.47·18-s + 4.10·19-s + 12.5·20-s + 21-s − 5.82·22-s + 9.09·23-s − 5.27·24-s + 4.16·25-s + 5.79·26-s − 27-s − 4.13·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 0.577·3-s + 2.06·4-s + 1.35·5-s − 1.01·6-s − 0.377·7-s + 1.86·8-s + 0.333·9-s + 2.37·10-s − 0.709·11-s − 1.19·12-s + 0.649·13-s − 0.661·14-s − 0.781·15-s + 1.20·16-s − 0.547·17-s + 0.583·18-s + 0.942·19-s + 2.79·20-s + 0.218·21-s − 1.24·22-s + 1.89·23-s − 1.07·24-s + 0.833·25-s + 1.13·26-s − 0.192·27-s − 0.780·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.947845840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.947845840\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 5 | \( 1 - 3.02T + 5T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 - 4.10T + 19T^{2} \) |
| 23 | \( 1 - 9.09T + 23T^{2} \) |
| 29 | \( 1 - 7.05T + 29T^{2} \) |
| 31 | \( 1 + 0.186T + 31T^{2} \) |
| 37 | \( 1 - 4.21T + 37T^{2} \) |
| 41 | \( 1 - 5.53T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 + 9.49T + 47T^{2} \) |
| 53 | \( 1 + 6.31T + 53T^{2} \) |
| 59 | \( 1 - 4.49T + 59T^{2} \) |
| 61 | \( 1 - 5.52T + 61T^{2} \) |
| 67 | \( 1 - 5.69T + 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 - 1.48T + 73T^{2} \) |
| 79 | \( 1 - 1.32T + 79T^{2} \) |
| 83 | \( 1 - 0.443T + 83T^{2} \) |
| 89 | \( 1 - 1.55T + 89T^{2} \) |
| 97 | \( 1 - 9.08T + 97T^{2} \) |
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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
−7.37181833381690827931776527890, −6.53643553258713366147073795197, −6.41683263075844633071803928994, −5.56837476634167151882762625520, −5.07146910390832410243012363464, −4.65098554462201760480420773677, −3.45101654305540980637211624537, −2.90419799584193655771483543513, −2.09802814110157390341596103607, −1.08214505006058680927314961210,
1.08214505006058680927314961210, 2.09802814110157390341596103607, 2.90419799584193655771483543513, 3.45101654305540980637211624537, 4.65098554462201760480420773677, 5.07146910390832410243012363464, 5.56837476634167151882762625520, 6.41683263075844633071803928994, 6.53643553258713366147073795197, 7.37181833381690827931776527890
