| L(s) = 1 | + 5.54·3-s − 6.21·5-s + 10.0·7-s + 3.69·9-s − 19.8·13-s − 34.4·15-s − 0.0283·17-s − 72.4·19-s + 55.9·21-s + 92.3·23-s − 86.3·25-s − 129.·27-s − 83.3·29-s − 172.·31-s − 62.7·35-s + 178.·37-s − 109.·39-s + 235.·41-s + 69.3·43-s − 22.9·45-s − 433.·47-s − 241.·49-s − 0.157·51-s − 599.·53-s − 401.·57-s − 365.·59-s + 685.·61-s + ⋯ |
| L(s) = 1 | + 1.06·3-s − 0.556·5-s + 0.545·7-s + 0.136·9-s − 0.423·13-s − 0.592·15-s − 0.000404·17-s − 0.874·19-s + 0.581·21-s + 0.837·23-s − 0.690·25-s − 0.920·27-s − 0.533·29-s − 0.996·31-s − 0.303·35-s + 0.792·37-s − 0.451·39-s + 0.897·41-s + 0.246·43-s − 0.0760·45-s − 1.34·47-s − 0.702·49-s − 0.000431·51-s − 1.55·53-s − 0.932·57-s − 0.805·59-s + 1.43·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 5.54T + 27T^{2} \) |
| 5 | \( 1 + 6.21T + 125T^{2} \) |
| 7 | \( 1 - 10.0T + 343T^{2} \) |
| 13 | \( 1 + 19.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 0.0283T + 4.91e3T^{2} \) |
| 19 | \( 1 + 72.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 92.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 83.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 172.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 178.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 235.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 69.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 433.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 599.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 365.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 685.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 224.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 254.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 882.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 386.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.23e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 891.T + 9.12e5T^{2} \) |
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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
−9.159135760380229778999496810148, −8.227522608899253326586919064778, −7.83820701968277111089247311924, −6.91140773499452338568734461908, −5.69203489861138295162868686948, −4.58682009742061745453409797941, −3.71370593287707746694173936801, −2.72726289419171629133252144353, −1.71157330777752363401159457228, 0,
1.71157330777752363401159457228, 2.72726289419171629133252144353, 3.71370593287707746694173936801, 4.58682009742061745453409797941, 5.69203489861138295162868686948, 6.91140773499452338568734461908, 7.83820701968277111089247311924, 8.227522608899253326586919064778, 9.159135760380229778999496810148
