| L(s) = 1 | + (−2.90 − 0.751i)3-s + (−1.38 + 4.80i)5-s + 3.03i·7-s + (7.86 + 4.36i)9-s − 11.4·11-s + 16.9·13-s + (7.63 − 12.9i)15-s + 18.1·17-s + 16.9i·19-s + (2.28 − 8.82i)21-s + 22.3·23-s + (−21.1 − 13.3i)25-s + (−19.5 − 18.5i)27-s − 17.8·29-s + 16.9·31-s + ⋯ |
| L(s) = 1 | + (−0.968 − 0.250i)3-s + (−0.277 + 0.960i)5-s + 0.434i·7-s + (0.874 + 0.485i)9-s − 1.04·11-s + 1.30·13-s + (0.509 − 0.860i)15-s + 1.06·17-s + 0.891i·19-s + (0.108 − 0.420i)21-s + 0.971·23-s + (−0.846 − 0.532i)25-s + (−0.724 − 0.688i)27-s − 0.617·29-s + 0.547·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8634325279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8634325279\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.90 + 0.751i)T \) |
| 5 | \( 1 + (1.38 - 4.80i)T \) |
| good | 7 | \( 1 - 3.03iT - 49T^{2} \) |
| 11 | \( 1 + 11.4T + 121T^{2} \) |
| 13 | \( 1 - 16.9T + 169T^{2} \) |
| 17 | \( 1 - 18.1T + 289T^{2} \) |
| 19 | \( 1 - 16.9iT - 361T^{2} \) |
| 23 | \( 1 - 22.3T + 529T^{2} \) |
| 29 | \( 1 + 17.8T + 841T^{2} \) |
| 31 | \( 1 - 16.9T + 961T^{2} \) |
| 37 | \( 1 - 1.21T + 1.36e3T^{2} \) |
| 41 | \( 1 - 42.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.98T + 1.84e3T^{2} \) |
| 47 | \( 1 + 52.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 8.58iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 93.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 80.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 38.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 109. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 28.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 56.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 126. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 153. iT - 9.40e3T^{2} \) |
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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
−10.47040041770114285055733784671, −9.532676977856000687183197449332, −8.145981928661181382439333890506, −7.64691436757390573063967465060, −6.58347764042176801023169386487, −5.91417151845181646713698602913, −5.15577201441922503578214675215, −3.82892482802893938161591268721, −2.79505569658477448505120974030, −1.31989584896450607160543147721,
0.36504962522416643861668710520, 1.35176899718471015508275415992, 3.32043976662192335914646353421, 4.34839257887077574767445257837, 5.18556620041558024754499046915, 5.81459615828592977127041377321, 6.96167913404080894576826444847, 7.82201565309299512188047788020, 8.719972120989523233271335963666, 9.602397096543389184852217075806
