| L(s) = 1 | − 2.69·2-s + 5.27·4-s + 3.42·5-s − 2.44·7-s − 8.83·8-s − 9.22·10-s + 4.40·11-s + 4.26·13-s + 6.58·14-s + 13.2·16-s − 3.71·17-s + 1.44·19-s + 18.0·20-s − 11.8·22-s − 2.28·23-s + 6.70·25-s − 11.4·26-s − 12.8·28-s + 2.25·29-s − 18.1·32-s + 10.0·34-s − 8.35·35-s − 2.58·37-s − 3.89·38-s − 30.2·40-s + 4.77·41-s + 5.45·43-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.63·4-s + 1.52·5-s − 0.923·7-s − 3.12·8-s − 2.91·10-s + 1.32·11-s + 1.18·13-s + 1.76·14-s + 3.32·16-s − 0.900·17-s + 0.330·19-s + 4.03·20-s − 2.53·22-s − 0.476·23-s + 1.34·25-s − 2.25·26-s − 2.43·28-s + 0.418·29-s − 3.21·32-s + 1.71·34-s − 1.41·35-s − 0.425·37-s − 0.631·38-s − 4.78·40-s + 0.745·41-s + 0.832·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.227878081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.227878081\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 - 3.42T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 4.40T + 11T^{2} \) |
| 13 | \( 1 - 4.26T + 13T^{2} \) |
| 17 | \( 1 + 3.71T + 17T^{2} \) |
| 19 | \( 1 - 1.44T + 19T^{2} \) |
| 23 | \( 1 + 2.28T + 23T^{2} \) |
| 29 | \( 1 - 2.25T + 29T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 - 4.77T + 41T^{2} \) |
| 43 | \( 1 - 5.45T + 43T^{2} \) |
| 47 | \( 1 - 0.225T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 - 3.56T + 59T^{2} \) |
| 61 | \( 1 - 9.41T + 61T^{2} \) |
| 67 | \( 1 - 8.78T + 67T^{2} \) |
| 71 | \( 1 + 2.28T + 71T^{2} \) |
| 73 | \( 1 - 6.09T + 73T^{2} \) |
| 79 | \( 1 - 16.8T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−8.021017683657018699042525361374, −6.94834855322915167561731282439, −6.46122172708323873550475124631, −6.26210384121154964537696526917, −5.48279356440495653170283180177, −3.97896002908704005376873018072, −3.06813210599808634452672531831, −2.21025130361003162372731722018, −1.54516246807806961394104149611, −0.75990716283624966113767262898,
0.75990716283624966113767262898, 1.54516246807806961394104149611, 2.21025130361003162372731722018, 3.06813210599808634452672531831, 3.97896002908704005376873018072, 5.48279356440495653170283180177, 6.26210384121154964537696526917, 6.46122172708323873550475124631, 6.94834855322915167561731282439, 8.021017683657018699042525361374
