L(s) = 1 | + (1.16 + 2.57i)2-s + (−5.30 + 5.98i)4-s − 16.7i·5-s + 19.3i·7-s + (−21.5 − 6.74i)8-s + (43.1 − 19.4i)10-s − 4.88·11-s − 12.0·13-s + (−49.7 + 22.3i)14-s + (−7.64 − 63.5i)16-s − 71.2i·17-s − 68.3i·19-s + (100. + 88.8i)20-s + (−5.66 − 12.5i)22-s + 136.·23-s + ⋯ |
L(s) = 1 | + (0.410 + 0.912i)2-s + (−0.663 + 0.748i)4-s − 1.49i·5-s + 1.04i·7-s + (−0.954 − 0.298i)8-s + (1.36 − 0.613i)10-s − 0.133·11-s − 0.257·13-s + (−0.950 + 0.427i)14-s + (−0.119 − 0.992i)16-s − 1.01i·17-s − 0.824i·19-s + (1.11 + 0.993i)20-s + (−0.0548 − 0.122i)22-s + 1.23·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.439709181\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.439709181\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.16 - 2.57i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 16.7iT - 125T^{2} \) |
| 7 | \( 1 - 19.3iT - 343T^{2} \) |
| 11 | \( 1 + 4.88T + 1.33e3T^{2} \) |
| 13 | \( 1 + 12.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 68.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 219. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 329. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 34.1iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 0.644iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 186.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 266. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 208.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 1.60T + 2.26e5T^{2} \) |
| 67 | \( 1 - 428. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 386.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 776.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 79.0iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 925.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.46e3T + 9.12e5T^{2} \) |
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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
−11.55923398495198197209900742742, −9.551536883839080407005288719525, −9.067590895369872296025284266734, −8.272782928453937003850199007891, −7.25643978198557532614605766446, −5.89875811142801455837666278396, −5.14107234501113055909023233094, −4.38260656841917081588744603584, −2.63821028474659719918125527594, −0.46816308184580926995506014128,
1.49818136120132220082503357197, 3.04526849228951043310862563163, 3.73248508771212670413528387006, 5.10543196534326905519255262750, 6.47395629425784733738228923519, 7.23095941266747075773742951237, 8.605825103223543533395234084187, 9.963218970228532172980728616898, 10.66120603908259081499333725331, 10.90295961398094489231278624141