| L(s) = 1 | − 0.528·2-s + 0.931·3-s − 1.72·4-s + 0.425·5-s − 0.492·6-s + 2.15·7-s + 1.96·8-s − 2.13·9-s − 0.225·10-s − 2.72·11-s − 1.60·12-s + 4.70·13-s − 1.14·14-s + 0.396·15-s + 2.40·16-s − 1.13·17-s + 1.12·18-s + 4.22·19-s − 0.732·20-s + 2.01·21-s + 1.43·22-s − 0.880·23-s + 1.83·24-s − 4.81·25-s − 2.48·26-s − 4.78·27-s − 3.71·28-s + ⋯ |
| L(s) = 1 | − 0.373·2-s + 0.537·3-s − 0.860·4-s + 0.190·5-s − 0.201·6-s + 0.815·7-s + 0.695·8-s − 0.710·9-s − 0.0711·10-s − 0.820·11-s − 0.462·12-s + 1.30·13-s − 0.304·14-s + 0.102·15-s + 0.600·16-s − 0.274·17-s + 0.265·18-s + 0.970·19-s − 0.163·20-s + 0.438·21-s + 0.306·22-s − 0.183·23-s + 0.374·24-s − 0.963·25-s − 0.488·26-s − 0.920·27-s − 0.701·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.501207664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.501207664\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 0.528T + 2T^{2} \) |
| 3 | \( 1 - 0.931T + 3T^{2} \) |
| 5 | \( 1 - 0.425T + 5T^{2} \) |
| 7 | \( 1 - 2.15T + 7T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 - 4.70T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 - 4.22T + 19T^{2} \) |
| 23 | \( 1 + 0.880T + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 + 0.386T + 31T^{2} \) |
| 37 | \( 1 + 7.52T + 37T^{2} \) |
| 41 | \( 1 - 2.51T + 41T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 5.62T + 73T^{2} \) |
| 79 | \( 1 + 1.39T + 79T^{2} \) |
| 83 | \( 1 + 1.35T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 5.81T + 97T^{2} \) |
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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
−8.990852286660281917335073311715, −8.418430454562712584224435789496, −8.116555527441896597494271554309, −7.14495813400069314664222401833, −5.78997514164072368273099924654, −5.27550091722449258444520779642, −4.24010354825440662907539188512, −3.35081420495994166266525485750, −2.18391524852273673337965979081, −0.890485018059039997946216016537,
0.890485018059039997946216016537, 2.18391524852273673337965979081, 3.35081420495994166266525485750, 4.24010354825440662907539188512, 5.27550091722449258444520779642, 5.78997514164072368273099924654, 7.14495813400069314664222401833, 8.116555527441896597494271554309, 8.418430454562712584224435789496, 8.990852286660281917335073311715
