| L(s) = 1 | − 1.41·2-s − 4.61·3-s + 2.00·4-s + 3.27i·5-s + 6.52·6-s − 2.64i·7-s − 2.82·8-s + 12.3·9-s − 4.63i·10-s + 6.52i·11-s − 9.23·12-s + 17.0·13-s + 3.74i·14-s − 15.1i·15-s + 4.00·16-s − 29.2i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.53·3-s + 0.500·4-s + 0.655i·5-s + 1.08·6-s − 0.377i·7-s − 0.353·8-s + 1.36·9-s − 0.463i·10-s + 0.593i·11-s − 0.769·12-s + 1.31·13-s + 0.267i·14-s − 1.00i·15-s + 0.250·16-s − 1.72i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0499595 + 0.232467i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0499595 + 0.232467i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 7 | \( 1 + 2.64iT \) |
| 23 | \( 1 + (9.44 - 20.9i)T \) |
| good | 3 | \( 1 + 4.61T + 9T^{2} \) |
| 5 | \( 1 - 3.27iT - 25T^{2} \) |
| 11 | \( 1 - 6.52iT - 121T^{2} \) |
| 13 | \( 1 - 17.0T + 169T^{2} \) |
| 17 | \( 1 + 29.2iT - 289T^{2} \) |
| 19 | \( 1 - 18.9iT - 361T^{2} \) |
| 29 | \( 1 + 39.4T + 841T^{2} \) |
| 31 | \( 1 + 53.1T + 961T^{2} \) |
| 37 | \( 1 - 14.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 54.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 68.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 20.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 33.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 5.15T + 3.48e3T^{2} \) |
| 61 | \( 1 + 60.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 85.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 34.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 0.238T + 5.32e3T^{2} \) |
| 79 | \( 1 - 108. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 158. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 143. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 22.2iT - 9.40e3T^{2} \) |
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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
−11.33807837985154890602443380508, −11.08542646051994298673792603101, −10.12515642905402609162767848203, −9.298558058217707929641902680394, −7.72637763563357759366280781418, −6.94980330117643248963802422789, −6.10225063275746523534567939426, −5.10185466541721555231111321620, −3.52460673484995044633605042009, −1.45543444228011610271405500416,
0.18705728797109664746345287032, 1.56560573786808069690163985158, 3.87039816882539597210609718911, 5.37346915964199496509194910379, 5.99671740756794694849361424058, 6.92946767570934312752281734994, 8.450959115104527150787108650828, 8.947700000868169927756584880887, 10.43454928551928803346375382505, 10.93595910304155691903845382575
