| L(s) = 1 | − 0.750·3-s − 0.872·7-s − 2.43·9-s − 1.11·11-s − 4.58·13-s + 2.96·17-s + 19-s + 0.654·21-s − 3.83·23-s + 4.07·27-s + 7.56·29-s + 9.56·31-s + 0.832·33-s − 0.750·37-s + 3.43·39-s + 8.87·41-s + 11.5·43-s − 8.53·47-s − 6.23·49-s − 2.22·51-s − 8.17·53-s − 0.750·57-s + 4.65·59-s + 0.889·61-s + 2.12·63-s − 2.59·67-s + 2.87·69-s + ⋯ |
| L(s) = 1 | − 0.433·3-s − 0.329·7-s − 0.812·9-s − 0.334·11-s − 1.27·13-s + 0.717·17-s + 0.229·19-s + 0.142·21-s − 0.799·23-s + 0.784·27-s + 1.40·29-s + 1.71·31-s + 0.144·33-s − 0.123·37-s + 0.550·39-s + 1.38·41-s + 1.75·43-s − 1.24·47-s − 0.891·49-s − 0.310·51-s − 1.12·53-s − 0.0993·57-s + 0.605·59-s + 0.113·61-s + 0.268·63-s − 0.316·67-s + 0.346·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.069859181\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.069859181\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.750T + 3T^{2} \) |
| 7 | \( 1 + 0.872T + 7T^{2} \) |
| 11 | \( 1 + 1.11T + 11T^{2} \) |
| 13 | \( 1 + 4.58T + 13T^{2} \) |
| 17 | \( 1 - 2.96T + 17T^{2} \) |
| 23 | \( 1 + 3.83T + 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 + 0.750T + 37T^{2} \) |
| 41 | \( 1 - 8.87T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 + 8.53T + 47T^{2} \) |
| 53 | \( 1 + 8.17T + 53T^{2} \) |
| 59 | \( 1 - 4.65T + 59T^{2} \) |
| 61 | \( 1 - 0.889T + 61T^{2} \) |
| 67 | \( 1 + 2.59T + 67T^{2} \) |
| 71 | \( 1 - 7.34T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 3.83T + 83T^{2} \) |
| 89 | \( 1 - 1.34T + 89T^{2} \) |
| 97 | \( 1 + 8.17T + 97T^{2} \) |
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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
−9.423407520489386876278378691600, −8.223210037870443634319751780392, −7.81278457547149526470197316889, −6.69296852693287183690595733801, −6.04713906802971174259037331141, −5.18428777430704076802784457896, −4.47230674440475733427138606442, −3.13215551145543620606969761439, −2.43139520037091939997664557467, −0.68904144651399685866559148777,
0.68904144651399685866559148777, 2.43139520037091939997664557467, 3.13215551145543620606969761439, 4.47230674440475733427138606442, 5.18428777430704076802784457896, 6.04713906802971174259037331141, 6.69296852693287183690595733801, 7.81278457547149526470197316889, 8.223210037870443634319751780392, 9.423407520489386876278378691600
