| L(s) = 1 | + 15.5·3-s − 96.7·5-s + 49·7-s − 0.445·9-s + 281.·11-s + 269.·13-s − 1.50e3·15-s + 1.71e3·17-s − 1.17e3·19-s + 763.·21-s − 785.·23-s + 6.23e3·25-s − 3.79e3·27-s − 6.14e3·29-s + 7.00e3·31-s + 4.38e3·33-s − 4.73e3·35-s + 1.14e4·37-s + 4.19e3·39-s − 1.32e4·41-s − 1.98e4·43-s + 43.0·45-s − 6.88e3·47-s + 2.40e3·49-s + 2.67e4·51-s − 8.65e3·53-s − 2.72e4·55-s + ⋯ |
| L(s) = 1 | + 0.999·3-s − 1.73·5-s + 0.377·7-s − 0.00183·9-s + 0.701·11-s + 0.442·13-s − 1.72·15-s + 1.44·17-s − 0.745·19-s + 0.377·21-s − 0.309·23-s + 1.99·25-s − 1.00·27-s − 1.35·29-s + 1.30·31-s + 0.700·33-s − 0.653·35-s + 1.38·37-s + 0.441·39-s − 1.23·41-s − 1.63·43-s + 0.00316·45-s − 0.454·47-s + 0.142·49-s + 1.44·51-s − 0.423·53-s − 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 15.5T + 243T^{2} \) |
| 5 | \( 1 + 96.7T + 3.12e3T^{2} \) |
| 11 | \( 1 - 281.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 269.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.71e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.17e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 785.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.14e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.32e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.98e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.88e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.78e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.07e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.64e4T + 8.58e9T^{2} \) |
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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
−9.662997700123196625103277743056, −8.632133469694280567934644288930, −8.055293378492272352878430780646, −7.51149085407284730129050631782, −6.20612156606446715442394197321, −4.66289318613400213650773080573, −3.71896435759485326372570724066, −3.09740284118574835355016953005, −1.46051276429475999513353927990, 0,
1.46051276429475999513353927990, 3.09740284118574835355016953005, 3.71896435759485326372570724066, 4.66289318613400213650773080573, 6.20612156606446715442394197321, 7.51149085407284730129050631782, 8.055293378492272352878430780646, 8.632133469694280567934644288930, 9.662997700123196625103277743056
