| L(s) = 1 | + 3.41i·3-s + 0.558·7-s − 2.66·9-s + (2.82 + 10.6i)11-s − 18.1·13-s + 32.0·17-s + 15.7i·19-s + 1.90i·21-s − 31.7i·23-s + 21.6i·27-s + 48.3i·29-s − 43.3·31-s + (−36.3 + 9.63i)33-s + 21.4i·37-s − 61.8i·39-s + ⋯ |
| L(s) = 1 | + 1.13i·3-s + 0.0798·7-s − 0.296·9-s + (0.256 + 0.966i)11-s − 1.39·13-s + 1.88·17-s + 0.828i·19-s + 0.0908i·21-s − 1.38i·23-s + 0.801i·27-s + 1.66i·29-s − 1.39·31-s + (−1.10 + 0.291i)33-s + 0.579i·37-s − 1.58i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.353112721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.353112721\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-2.82 - 10.6i)T \) |
| good | 3 | \( 1 - 3.41iT - 9T^{2} \) |
| 7 | \( 1 - 0.558T + 49T^{2} \) |
| 13 | \( 1 + 18.1T + 169T^{2} \) |
| 17 | \( 1 - 32.0T + 289T^{2} \) |
| 19 | \( 1 - 15.7iT - 361T^{2} \) |
| 23 | \( 1 + 31.7iT - 529T^{2} \) |
| 29 | \( 1 - 48.3iT - 841T^{2} \) |
| 31 | \( 1 + 43.3T + 961T^{2} \) |
| 37 | \( 1 - 21.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 38.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 75.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 8.43iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 29.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 64.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 18.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 94.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 0.945T + 5.32e3T^{2} \) |
| 79 | \( 1 + 73.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 161.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 91.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + 33.1iT - 9.40e3T^{2} \) |
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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
−10.06874810377132919884680366510, −9.532712278423615287302649858380, −8.534709042953256689720854962063, −7.52681114231465763011960475239, −6.83652309004717912526134264788, −5.40486970924983987864646711516, −4.91471769472086975670697524814, −3.96554365004116867810895744041, −3.03794693189949032990429721005, −1.60054666570514373308875239878,
0.41310965870806992420544852383, 1.55570299799396207042692713991, 2.71010819357699158534808992058, 3.79911874819644858726568289427, 5.21645455305739234649234796978, 5.89127609654718786897691890618, 6.95228645850856867302920654883, 7.62075083740675124693167039704, 8.101990277590847821514304633364, 9.405425585383900337343459652384
