| L(s) = 1 | − 124.·7-s − 80·11-s + 374.·13-s + 546.·17-s − 12·19-s − 2.03e3·23-s + 4.56e3·29-s − 344·31-s + 4.37e3·37-s − 1.42e4·41-s + 1.89e4·43-s + 2.45e4·47-s − 1.19e3·49-s − 2.74e4·53-s + 3.80e4·59-s − 8.20e3·61-s + 1.32e4·67-s − 4.84e4·71-s − 4.24e4·73-s + 9.99e3·77-s − 9.26e3·79-s − 3.34e4·83-s − 2.43e4·89-s − 4.68e4·91-s − 1.36e5·97-s + 1.05e5·101-s − 6.81e4·103-s + ⋯ |
| L(s) = 1 | − 0.963·7-s − 0.199·11-s + 0.615·13-s + 0.458·17-s − 0.00762·19-s − 0.800·23-s + 1.00·29-s − 0.0642·31-s + 0.525·37-s − 1.32·41-s + 1.56·43-s + 1.61·47-s − 0.0708·49-s − 1.34·53-s + 1.42·59-s − 0.282·61-s + 0.360·67-s − 1.14·71-s − 0.933·73-s + 0.192·77-s − 0.167·79-s − 0.532·83-s − 0.325·89-s − 0.593·91-s − 1.47·97-s + 1.02·101-s − 0.632·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 124.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 80T + 1.61e5T^{2} \) |
| 13 | \( 1 - 374.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 546.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 12T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.03e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.56e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 344T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.37e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.42e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.89e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.45e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.74e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.80e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.20e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.26e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.34e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.43e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.36e5T + 8.58e9T^{2} \) |
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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
−8.971885254172947948982856524211, −8.151200696081706078717972432446, −7.20430956300476274579830956564, −6.29427373750045940846621986846, −5.63779247485808698747439641218, −4.38405931791744144937798433514, −3.45241365248239048368801128223, −2.53047997087968762562723473511, −1.15720636833634770245908925876, 0,
1.15720636833634770245908925876, 2.53047997087968762562723473511, 3.45241365248239048368801128223, 4.38405931791744144937798433514, 5.63779247485808698747439641218, 6.29427373750045940846621986846, 7.20430956300476274579830956564, 8.151200696081706078717972432446, 8.971885254172947948982856524211
