L(s) = 1 | + 1.41·2-s + 0.717i·3-s + 2.00·4-s + 6.63i·5-s + 1.01i·6-s + 2.82·8-s + 8.48·9-s + 9.37i·10-s − 4.75·11-s + 1.43i·12-s − 15.2i·13-s − 4.75·15-s + 4.00·16-s − 3.76i·17-s + 12·18-s − 4.18i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.239i·3-s + 0.500·4-s + 1.32i·5-s + 0.169i·6-s + 0.353·8-s + 0.942·9-s + 0.937i·10-s − 0.432·11-s + 0.119i·12-s − 1.17i·13-s − 0.317·15-s + 0.250·16-s − 0.221i·17-s + 0.666·18-s − 0.220i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.85180 + 0.690398i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.85180 + 0.690398i\) |
\(L(2)\) |
|
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 | \( 1 \) |
good | 3 | \( 1 - 0.717iT - 9T^{2} \) |
| 5 | \( 1 - 6.63iT - 25T^{2} \) |
| 11 | \( 1 + 4.75T + 121T^{2} \) |
| 13 | \( 1 + 15.2iT - 169T^{2} \) |
| 17 | \( 1 + 3.76iT - 289T^{2} \) |
| 19 | \( 1 + 4.18iT - 361T^{2} \) |
| 23 | \( 1 + 27.7T + 529T^{2} \) |
| 29 | \( 1 - 3.51T + 841T^{2} \) |
| 31 | \( 1 + 48.8iT - 961T^{2} \) |
| 37 | \( 1 + 2.94T + 1.36e3T^{2} \) |
| 41 | \( 1 - 27.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 10.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 52.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 55.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 38.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 90.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 34.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 36.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 52.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 33.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 127. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 50.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 101. iT - 9.40e3T^{2} \) |
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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
−13.77402522047557938585766041221, −12.92972884516425483358802108347, −11.67189013892563541958796166674, −10.55241010020832196862253095236, −9.963814611820126669969414253910, −7.87603456012005304510217729510, −6.89902441258110343322973441282, −5.63115656179015135495985012503, −4.00685145887804941586479636368, −2.64393057434298362704356319025,
1.67968749432020468768486290086, 4.08898896463968764285484657700, 5.08541992710535575449551569317, 6.55976036099246848100284737732, 7.894663502334751161153183135360, 9.138492025446357297643928110634, 10.39833160818320175721954226876, 11.93885635854444153528076922099, 12.54103316394758386138211565945, 13.41623354791291355816044803837