L(s) = 1 | + 5.13·3-s + 6.19i·7-s + 17.3·9-s + 20.0·11-s − 15.8i·13-s − 6.98·17-s + 10.3·19-s + 31.7i·21-s + 22.3i·23-s + 42.9·27-s + 4.20i·29-s + 20.7i·31-s + 102.·33-s − 35.4i·37-s − 81.4i·39-s + ⋯ |
L(s) = 1 | + 1.71·3-s + 0.884i·7-s + 1.92·9-s + 1.82·11-s − 1.22i·13-s − 0.411·17-s + 0.546·19-s + 1.51i·21-s + 0.973i·23-s + 1.58·27-s + 0.145i·29-s + 0.668i·31-s + 3.11·33-s − 0.958i·37-s − 2.08i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(4.457509908\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.457509908\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 5.13T + 9T^{2} \) |
| 7 | \( 1 - 6.19iT - 49T^{2} \) |
| 11 | \( 1 - 20.0T + 121T^{2} \) |
| 13 | \( 1 + 15.8iT - 169T^{2} \) |
| 17 | \( 1 + 6.98T + 289T^{2} \) |
| 19 | \( 1 - 10.3T + 361T^{2} \) |
| 23 | \( 1 - 22.3iT - 529T^{2} \) |
| 29 | \( 1 - 4.20iT - 841T^{2} \) |
| 31 | \( 1 - 20.7iT - 961T^{2} \) |
| 37 | \( 1 + 35.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 37.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 23.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 48.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 77.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 0.497T + 3.48e3T^{2} \) |
| 61 | \( 1 + 60.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 82.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 28.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 87.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 103.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 49.2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 84.4T + 9.40e3T^{2} \) |
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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
−9.227964417380835552663124032973, −8.589108680158156895061155718498, −7.85210740586304564396496211798, −7.06824536665584796256484340307, −6.10387608693656407419679164968, −5.02501613242954885229052917538, −3.77838994166068184189675868788, −3.28904804251018318823554184784, −2.26653083224549884046361996387, −1.28829389660153773229885929668,
1.17407427965998193290095653079, 2.06600174538231840334716251449, 3.23315277508095067907011822349, 4.11333628339965820547572093171, 4.42886544214653271267121243347, 6.31333980349927304616171424111, 6.97903871955057377703433505077, 7.58249429973751243694730972344, 8.680083563113810458706728302965, 9.002574201646519006753503715107