| L(s) = 1 | + 1.71·3-s + 31.3·7-s − 24.0·9-s + 64.2·11-s − 13·13-s − 131.·17-s − 77.0·19-s + 53.8·21-s − 192.·23-s − 87.5·27-s − 162.·29-s − 205.·31-s + 110.·33-s − 271.·37-s − 22.2·39-s + 441.·41-s − 3.15·43-s − 106.·47-s + 641.·49-s − 226.·51-s − 63.1·53-s − 132.·57-s + 31.9·59-s + 278.·61-s − 754.·63-s + 298.·67-s − 329.·69-s + ⋯ |
| L(s) = 1 | + 0.330·3-s + 1.69·7-s − 0.891·9-s + 1.76·11-s − 0.277·13-s − 1.88·17-s − 0.930·19-s + 0.559·21-s − 1.74·23-s − 0.624·27-s − 1.04·29-s − 1.19·31-s + 0.581·33-s − 1.20·37-s − 0.0915·39-s + 1.68·41-s − 0.0111·43-s − 0.330·47-s + 1.87·49-s − 0.621·51-s − 0.163·53-s − 0.307·57-s + 0.0704·59-s + 0.584·61-s − 1.50·63-s + 0.544·67-s − 0.574·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 - 1.71T + 27T^{2} \) |
| 7 | \( 1 - 31.3T + 343T^{2} \) |
| 11 | \( 1 - 64.2T + 1.33e3T^{2} \) |
| 17 | \( 1 + 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 162.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 205.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 271.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 441.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 3.15T + 7.95e4T^{2} \) |
| 47 | \( 1 + 106.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 63.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 31.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 278.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 298.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 931.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 107.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 354.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 974.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 232.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 965.T + 9.12e5T^{2} \) |
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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
−8.808632938973832161647603505615, −8.261143495775605019315080477304, −7.32968567108699606946192429498, −6.37756479246738278629915973144, −5.52220699803535105231134594974, −4.35647619933101088396683775603, −3.91112619350551806133422167813, −2.20906600521850189073298205847, −1.71003214824510173079896075151, 0,
1.71003214824510173079896075151, 2.20906600521850189073298205847, 3.91112619350551806133422167813, 4.35647619933101088396683775603, 5.52220699803535105231134594974, 6.37756479246738278629915973144, 7.32968567108699606946192429498, 8.261143495775605019315080477304, 8.808632938973832161647603505615
