L(s) = 1 | + 1.41i·2-s + (2.81 + 1.03i)3-s − 2.00·4-s + (6.63 − 3.83i)5-s + (−1.45 + 3.98i)6-s + (2.93 − 6.35i)7-s − 2.82i·8-s + (6.87 + 5.80i)9-s + (5.42 + 9.38i)10-s + (−8.62 − 4.98i)11-s + (−5.63 − 2.06i)12-s + (−5.46 + 9.46i)13-s + (8.98 + 4.15i)14-s + (22.6 − 3.95i)15-s + 4.00·16-s + (−24.5 + 14.1i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.939 + 0.343i)3-s − 0.500·4-s + (1.32 − 0.766i)5-s + (−0.242 + 0.664i)6-s + (0.419 − 0.907i)7-s − 0.353i·8-s + (0.763 + 0.645i)9-s + (0.542 + 0.938i)10-s + (−0.784 − 0.452i)11-s + (−0.469 − 0.171i)12-s + (−0.420 + 0.727i)13-s + (0.641 + 0.296i)14-s + (1.51 − 0.263i)15-s + 0.250·16-s + (−1.44 + 0.833i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.93879 + 0.679545i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.93879 + 0.679545i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (-2.81 - 1.03i)T \) |
| 7 | \( 1 + (-2.93 + 6.35i)T \) |
good | 5 | \( 1 + (-6.63 + 3.83i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (8.62 + 4.98i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (5.46 - 9.46i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (24.5 - 14.1i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (5.29 - 9.17i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (0.643 - 0.371i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (39.7 - 22.9i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 47.6T + 961T^{2} \) |
| 37 | \( 1 + (-12.6 + 21.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (32.2 + 18.6i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-11.0 - 19.1i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + 29.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-26.8 + 15.4i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + 58.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 25.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 1.60T + 4.48e3T^{2} \) |
| 71 | \( 1 - 98.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-43.7 - 75.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 - 107.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-135. + 78.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (54.2 + 31.2i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-30.3 - 52.6i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−13.47297957925351061315986627244, −12.90300973863794288023680246747, −10.77123643650284474407453393123, −9.849803637262751693225703339780, −8.902146892128510185179056118315, −8.070575689343287973429911334999, −6.71331804467454635548116065705, −5.23536243629375797551728859297, −4.16792243074472895315942618053, −1.94235162508570197513757166649,
2.22040270032356029088871793165, 2.68900844853857592976955820031, 4.91411536668824961054184249550, 6.38413973735342764101065567861, 7.80765018599989634087393503209, 9.078791108822563367862381499348, 9.798146891507645924946423884965, 10.81532809165672408551695125801, 12.13074864326174976073443733723, 13.35388293453364283990693162082