L(s) = 1 | − 3.86·5-s + 1.03·7-s − 5.46·11-s − 13-s + 5.65·17-s + 6.69·19-s + 6.92·23-s + 9.92·25-s − 7.72·29-s + 1.03·31-s − 3.99·35-s − 2·37-s + 3.86·41-s + 5.65·43-s − 9.46·47-s − 5.92·49-s + 5.65·53-s + 21.1·55-s − 2.53·59-s + 4.92·61-s + 3.86·65-s − 6.69·67-s − 9.46·71-s − 15.8·73-s − 5.65·77-s − 13.4·83-s − 21.8·85-s + ⋯ |
L(s) = 1 | − 1.72·5-s + 0.391·7-s − 1.64·11-s − 0.277·13-s + 1.37·17-s + 1.53·19-s + 1.44·23-s + 1.98·25-s − 1.43·29-s + 0.185·31-s − 0.676·35-s − 0.328·37-s + 0.603·41-s + 0.862·43-s − 1.38·47-s − 0.846·49-s + 0.777·53-s + 2.84·55-s − 0.330·59-s + 0.630·61-s + 0.479·65-s − 0.817·67-s − 1.12·71-s − 1.85·73-s − 0.644·77-s − 1.47·83-s − 2.37·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 3.86T + 5T^{2} \) |
| 7 | \( 1 - 1.03T + 7T^{2} \) |
| 11 | \( 1 + 5.46T + 11T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 - 6.69T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 7.72T + 29T^{2} \) |
| 31 | \( 1 - 1.03T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 3.86T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + 9.46T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 + 6.69T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 + 15.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 0.277T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.87042371765579681680949923107, −7.56053530941796547032035994224, −7.13267202497260509368347394290, −5.58125190245468768604389971850, −5.15533557875806585870276579479, −4.33286847456498842977679959348, −3.28265111206026687942925406470, −2.89430725889198642379537699677, −1.20051858746041992913675771770, 0,
1.20051858746041992913675771770, 2.89430725889198642379537699677, 3.28265111206026687942925406470, 4.33286847456498842977679959348, 5.15533557875806585870276579479, 5.58125190245468768604389971850, 7.13267202497260509368347394290, 7.56053530941796547032035994224, 7.87042371765579681680949923107