| L(s) = 1 | − 25·5-s + 61.7·7-s − 218.·11-s − 864.·13-s − 521.·17-s − 1.45e3·19-s − 4.92e3·23-s + 625·25-s + 5.66e3·29-s − 2.98e3·31-s − 1.54e3·35-s − 110.·37-s + 2.41e3·41-s + 1.35e4·43-s + 1.52e4·47-s − 1.29e4·49-s − 1.04e4·53-s + 5.45e3·55-s + 3.17e4·59-s − 2.80e4·61-s + 2.16e4·65-s − 1.78e4·67-s + 3.67e4·71-s − 4.97e4·73-s − 1.34e4·77-s + 8.19e4·79-s − 9.74e4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.476·7-s − 0.544·11-s − 1.41·13-s − 0.437·17-s − 0.925·19-s − 1.94·23-s + 0.200·25-s + 1.25·29-s − 0.557·31-s − 0.212·35-s − 0.0132·37-s + 0.224·41-s + 1.12·43-s + 1.00·47-s − 0.773·49-s − 0.513·53-s + 0.243·55-s + 1.18·59-s − 0.966·61-s + 0.634·65-s − 0.487·67-s + 0.865·71-s − 1.09·73-s − 0.259·77-s + 1.47·79-s − 1.55·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9751682872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9751682872\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 25T \) |
| good | 7 | \( 1 - 61.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 218.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 864.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 521.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.45e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.92e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.66e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.98e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 110.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.35e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.52e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.78e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.67e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.97e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.74e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.81e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.58e4T + 8.58e9T^{2} \) |
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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
−9.107035982932696779294762138359, −8.149839599418389511038569327749, −7.66535795887336683356050990195, −6.71525733074551654870431487595, −5.70942281304417575380547189069, −4.69785180096006452000372907832, −4.09742521654689651168480830288, −2.70514692304616837714073436936, −1.94149230544641305632542682824, −0.40236643327620184646604400456,
0.40236643327620184646604400456, 1.94149230544641305632542682824, 2.70514692304616837714073436936, 4.09742521654689651168480830288, 4.69785180096006452000372907832, 5.70942281304417575380547189069, 6.71525733074551654870431487595, 7.66535795887336683356050990195, 8.149839599418389511038569327749, 9.107035982932696779294762138359
