| L(s) = 1 | − 0.585·3-s + 5-s − 2.65·9-s + 5.24·11-s + 0.585·13-s − 0.585·15-s + 1.17·17-s + 19-s + 1.58·23-s − 4·25-s + 3.31·27-s − 6.58·29-s − 7.41·31-s − 3.07·33-s − 9.41·37-s − 0.343·39-s − 5.65·41-s + 5.24·43-s − 2.65·45-s − 7.24·47-s − 0.686·51-s − 2.58·53-s + 5.24·55-s − 0.585·57-s − 4.82·59-s + 5·61-s + 0.585·65-s + ⋯ |
| L(s) = 1 | − 0.338·3-s + 0.447·5-s − 0.885·9-s + 1.58·11-s + 0.162·13-s − 0.151·15-s + 0.284·17-s + 0.229·19-s + 0.330·23-s − 0.800·25-s + 0.637·27-s − 1.22·29-s − 1.33·31-s − 0.534·33-s − 1.54·37-s − 0.0549·39-s − 0.883·41-s + 0.799·43-s − 0.396·45-s − 1.05·47-s − 0.0961·51-s − 0.355·53-s + 0.706·55-s − 0.0775·57-s − 0.628·59-s + 0.640·61-s + 0.0726·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.585T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 5.24T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 31 | \( 1 + 7.41T + 31T^{2} \) |
| 37 | \( 1 + 9.41T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 + 7.24T + 47T^{2} \) |
| 53 | \( 1 + 2.58T + 53T^{2} \) |
| 59 | \( 1 + 4.82T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 8.82T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 4.89T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 5.65T + 97T^{2} \) |
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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
−7.41527283719016309116380487071, −6.79023476294846877948972954878, −6.04835503109843366501841077094, −5.59479467479376490997219687023, −4.85238533744613234613494300952, −3.74915080154850877758998628668, −3.35193826353892365349383471969, −2.07350470775797066717421463576, −1.35707129579348894507876788365, 0,
1.35707129579348894507876788365, 2.07350470775797066717421463576, 3.35193826353892365349383471969, 3.74915080154850877758998628668, 4.85238533744613234613494300952, 5.59479467479376490997219687023, 6.04835503109843366501841077094, 6.79023476294846877948972954878, 7.41527283719016309116380487071
