| L(s) = 1 | + 1.24·2-s − 0.445·3-s − 0.445·4-s − 0.554·5-s − 0.554·6-s − 1.35·7-s − 3.04·8-s − 2.80·9-s − 0.692·10-s − 2.15·11-s + 0.198·12-s + 1.58·13-s − 1.69·14-s + 0.246·15-s − 2.91·16-s + 2.49·17-s − 3.49·18-s + 5.80·19-s + 0.246·20-s + 0.603·21-s − 2.69·22-s + 5.09·23-s + 1.35·24-s − 4.69·25-s + 1.97·26-s + 2.58·27-s + 0.603·28-s + ⋯ |
| L(s) = 1 | + 0.881·2-s − 0.256·3-s − 0.222·4-s − 0.248·5-s − 0.226·6-s − 0.512·7-s − 1.07·8-s − 0.933·9-s − 0.218·10-s − 0.650·11-s + 0.0571·12-s + 0.438·13-s − 0.452·14-s + 0.0637·15-s − 0.727·16-s + 0.604·17-s − 0.823·18-s + 1.33·19-s + 0.0552·20-s + 0.131·21-s − 0.573·22-s + 1.06·23-s + 0.276·24-s − 0.938·25-s + 0.386·26-s + 0.496·27-s + 0.114·28-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
| \(L(1)\) |
\(\approx\) |
\(1.550463534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.550463534\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.24T + 2T^{2} \) |
| 3 | \( 1 + 0.445T + 3T^{2} \) |
| 5 | \( 1 + 0.554T + 5T^{2} \) |
| 7 | \( 1 + 1.35T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 - 2.49T + 17T^{2} \) |
| 19 | \( 1 - 5.80T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 7.62T + 31T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 3.07T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 + 8.09T + 61T^{2} \) |
| 67 | \( 1 - 7.38T + 67T^{2} \) |
| 71 | \( 1 - 1.68T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 + 2.08T + 79T^{2} \) |
| 83 | \( 1 - 0.917T + 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−9.384266441565883695122332770220, −8.338906641868561516316533859195, −7.77494686559317854283764393927, −6.49264706546436716596580184498, −5.90988370336443668837317880771, −5.16625062277484521195012928593, −4.43093079687699160127310534176, −3.17905533826100351543722360593, −2.91509849934404604267717942191, −0.73963652845832953910809386062,
0.73963652845832953910809386062, 2.91509849934404604267717942191, 3.17905533826100351543722360593, 4.43093079687699160127310534176, 5.16625062277484521195012928593, 5.90988370336443668837317880771, 6.49264706546436716596580184498, 7.77494686559317854283764393927, 8.338906641868561516316533859195, 9.384266441565883695122332770220
