| L(s) = 1 | + 0.445·2-s − 1.80·4-s + 2.80·5-s − 7-s − 1.69·8-s + 1.24·10-s + 1.80·11-s − 3.80·13-s − 0.445·14-s + 2.85·16-s − 0.801·17-s − 5.13·19-s − 5.04·20-s + 0.801·22-s − 2.35·23-s + 2.85·25-s − 1.69·26-s + 1.80·28-s − 10.2·29-s + 0.493·31-s + 4.65·32-s − 0.356·34-s − 2.80·35-s − 1.33·37-s − 2.28·38-s − 4.74·40-s + 6.87·41-s + ⋯ |
| L(s) = 1 | + 0.314·2-s − 0.900·4-s + 1.25·5-s − 0.377·7-s − 0.598·8-s + 0.394·10-s + 0.543·11-s − 1.05·13-s − 0.118·14-s + 0.712·16-s − 0.194·17-s − 1.17·19-s − 1.12·20-s + 0.170·22-s − 0.491·23-s + 0.570·25-s − 0.331·26-s + 0.340·28-s − 1.90·29-s + 0.0887·31-s + 0.822·32-s − 0.0612·34-s − 0.473·35-s − 0.219·37-s − 0.370·38-s − 0.749·40-s + 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 239 | \( 1 - T \) |
| good | 2 | \( 1 - 0.445T + 2T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 1.80T + 11T^{2} \) |
| 13 | \( 1 + 3.80T + 13T^{2} \) |
| 17 | \( 1 + 0.801T + 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 0.493T + 31T^{2} \) |
| 37 | \( 1 + 1.33T + 37T^{2} \) |
| 41 | \( 1 - 6.87T + 41T^{2} \) |
| 43 | \( 1 - 1.55T + 43T^{2} \) |
| 47 | \( 1 - 8.25T + 47T^{2} \) |
| 53 | \( 1 + 7.93T + 53T^{2} \) |
| 59 | \( 1 + 6.85T + 59T^{2} \) |
| 61 | \( 1 - 9.08T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 + 7.78T + 71T^{2} \) |
| 73 | \( 1 + 4.77T + 73T^{2} \) |
| 79 | \( 1 + 9.70T + 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−9.054840941403411463313077197190, −7.988142762459435150878690622798, −7.00009108054118473181098939530, −6.03604687800154913046018757692, −5.64496743717433048922660541833, −4.62182007952028377642131201396, −3.91481908656148017733663667591, −2.71268358931477533413185938563, −1.72904497240586177669426889615, 0,
1.72904497240586177669426889615, 2.71268358931477533413185938563, 3.91481908656148017733663667591, 4.62182007952028377642131201396, 5.64496743717433048922660541833, 6.03604687800154913046018757692, 7.00009108054118473181098939530, 7.988142762459435150878690622798, 9.054840941403411463313077197190
