L(s) = 1 | − 2.53·3-s + 6.47i·5-s − 1.41·7-s − 2.56·9-s + 4.80i·11-s − 12.8·13-s − 16.4i·15-s − 14.3·17-s + (16.9 + 8.56i)19-s + 3.60·21-s − 22.4·23-s − 16.9·25-s + 29.3·27-s − 29.2·29-s − 9.22i·31-s + ⋯ |
L(s) = 1 | − 0.845·3-s + 1.29i·5-s − 0.202·7-s − 0.284·9-s + 0.437i·11-s − 0.986·13-s − 1.09i·15-s − 0.845·17-s + (0.892 + 0.450i)19-s + 0.171·21-s − 0.975·23-s − 0.677·25-s + 1.08·27-s − 1.00·29-s − 0.297i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2762930267\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2762930267\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-16.9 - 8.56i)T \) |
good | 3 | \( 1 + 2.53T + 9T^{2} \) |
| 5 | \( 1 - 6.47iT - 25T^{2} \) |
| 7 | \( 1 + 1.41T + 49T^{2} \) |
| 11 | \( 1 - 4.80iT - 121T^{2} \) |
| 13 | \( 1 + 12.8T + 169T^{2} \) |
| 17 | \( 1 + 14.3T + 289T^{2} \) |
| 23 | \( 1 + 22.4T + 529T^{2} \) |
| 29 | \( 1 + 29.2T + 841T^{2} \) |
| 31 | \( 1 + 9.22iT - 961T^{2} \) |
| 37 | \( 1 + 23.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 7.12iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 9.54iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 10.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 94.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 68.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 51.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 6.63T + 4.48e3T^{2} \) |
| 71 | \( 1 - 44.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 19.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 63.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 111. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 60.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 136. iT - 9.40e3T^{2} \) |
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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
−9.722945756350225925526548631704, −8.542017320008979255063134591420, −7.36695050578985806530796468884, −6.95579299391044597814784427717, −6.01764454440077249527210268817, −5.33866931867066361202262585429, −4.17432186462201973930603263001, −3.04326355516136470272080358961, −2.09591478388435191105025183111, −0.12157248351025862285808065965,
0.816278798107865014221636635053, 2.29237131875447505723576340942, 3.73190186995020332533815014795, 4.92744433406893448073780955656, 5.27948459583407746630594021960, 6.21081161117450946684886654785, 7.16931380814825596915939265768, 8.192232134541357153562303624681, 8.941270806321183367432884470098, 9.603155920282300203754128999254