| L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 44.5i·5-s + 22.6i·8-s + 126.·10-s − 50.9i·11-s − 43.9·13-s + 64.0·16-s − 320. i·17-s + 565.·19-s − 356. i·20-s − 144.·22-s − 364. i·23-s − 1.36e3·25-s + 124. i·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 1.78i·5-s + 0.353i·8-s + 1.26·10-s − 0.421i·11-s − 0.259·13-s + 0.250·16-s − 1.10i·17-s + 1.56·19-s − 0.891i·20-s − 0.297·22-s − 0.689i·23-s − 2.17·25-s + 0.183i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.815164335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.815164335\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 44.5iT - 625T^{2} \) |
| 11 | \( 1 + 50.9iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 43.9T + 2.85e4T^{2} \) |
| 17 | \( 1 + 320. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 565.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 364. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 263. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 71.0T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.15e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 3.08e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.61e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.05e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.16e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 5.05e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 280.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 921.T + 2.01e7T^{2} \) |
| 71 | \( 1 + 7.86e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 2.26e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 3.17e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 3.10e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.67e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.45e4T + 8.85e7T^{2} \) |
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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
−9.711408223659810622130455673291, −8.885707368888932113219435210181, −7.54406403032704365555720293592, −7.11396993411713743084972915027, −6.01520243252842056183403715447, −5.02725731155908034716826315400, −3.62601003971422177890403148845, −3.01210955182277509452244842725, −2.16890464346543091723583440205, −0.57187434762008391122358607575,
0.78031739048957045513073554313, 1.71949123440869040485593288208, 3.58363503430963726138029239701, 4.58149026319138810643348099756, 5.26515933391943943123905912232, 5.98115850298076525226681259874, 7.27704721063047731339234929145, 7.996056909819068712024030245515, 8.735736083406105422740823274792, 9.459365417740530505070794052078
