| L(s) = 1 | − 1.73·3-s + (−2 + 4.58i)5-s − 3.46·7-s + 2.99·9-s − 15.8i·11-s − 9.16i·13-s + (3.46 − 7.93i)15-s + 9.16i·17-s − 31.7i·19-s + 5.99·21-s − 27.7·23-s + (−17 − 18.3i)25-s − 5.19·27-s − 8·29-s + 27.4i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (−0.400 + 0.916i)5-s − 0.494·7-s + 0.333·9-s − 1.44i·11-s − 0.705i·13-s + (0.230 − 0.529i)15-s + 0.539i·17-s − 1.67i·19-s + 0.285·21-s − 1.20·23-s + (−0.680 − 0.733i)25-s − 0.192·27-s − 0.275·29-s + 0.833i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.292886 - 0.447391i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.292886 - 0.447391i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (2 - 4.58i)T \) |
| good | 7 | \( 1 + 3.46T + 49T^{2} \) |
| 11 | \( 1 + 15.8iT - 121T^{2} \) |
| 13 | \( 1 + 9.16iT - 169T^{2} \) |
| 17 | \( 1 - 9.16iT - 289T^{2} \) |
| 19 | \( 1 + 31.7iT - 361T^{2} \) |
| 23 | \( 1 + 27.7T + 529T^{2} \) |
| 29 | \( 1 + 8T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 45.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 50T + 1.68e3T^{2} \) |
| 43 | \( 1 + 62.3T + 1.84e3T^{2} \) |
| 47 | \( 1 - 48.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 27.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 15.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 26T + 3.72e3T^{2} \) |
| 67 | \( 1 - 55.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 95.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 128. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 126. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 131.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 86T + 7.92e3T^{2} \) |
| 97 | \( 1 + 109. iT - 9.40e3T^{2} \) |
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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
−11.30958119557840693654305195354, −10.90218393993485216041601334458, −9.898905748412805261601665683184, −8.601960275856539373127998428097, −7.49246109314488562746666276733, −6.42307191133223129071561502552, −5.64009371386387508719230951587, −3.95676197980158678407653602038, −2.81529821858172308101933299975, −0.30179231113352374400280791126,
1.70813606476582332066516456832, 3.92551232536884664585227996651, 4.84132006420214184274432350303, 6.05610197200564358484132590882, 7.22125452367053712072624122484, 8.216481235622918693462660942337, 9.564721251734200799835927196942, 10.04969402225733042723743796465, 11.55477854909142786372110511884, 12.22058591180293357146910833381
