| L(s) = 1 | − 1.41·5-s − 8·7-s − 11.3·11-s + 8i·13-s − 12.7i·17-s − 32i·19-s − 33.9i·23-s − 23·25-s + 43.8·29-s + 40·31-s + 11.3·35-s − 26i·37-s + 66.4i·41-s + 16i·43-s − 11.3i·47-s + ⋯ |
| L(s) = 1 | − 0.282·5-s − 1.14·7-s − 1.02·11-s + 0.615i·13-s − 0.748i·17-s − 1.68i·19-s − 1.47i·23-s − 0.920·25-s + 1.51·29-s + 1.29·31-s + 0.323·35-s − 0.702i·37-s + 1.62i·41-s + 0.372i·43-s − 0.240i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6527473561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6527473561\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.41T + 25T^{2} \) |
| 7 | \( 1 + 8T + 49T^{2} \) |
| 11 | \( 1 + 11.3T + 121T^{2} \) |
| 13 | \( 1 - 8iT - 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 + 32iT - 361T^{2} \) |
| 23 | \( 1 + 33.9iT - 529T^{2} \) |
| 29 | \( 1 - 43.8T + 841T^{2} \) |
| 31 | \( 1 - 40T + 961T^{2} \) |
| 37 | \( 1 + 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 66.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 11.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 32.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 22.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 54iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 80iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 79.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 104T + 6.24e3T^{2} \) |
| 83 | \( 1 + 101.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 77.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 80T + 9.40e3T^{2} \) |
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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
−8.944278499296394362993241624731, −8.326122163300055193096714778621, −7.33804371451333725940538294357, −6.68668230022550861493343757827, −6.05070379627125828743973424398, −4.82685441565450504527570724668, −4.34048504619227800822847280945, −2.85974755672689855418563153033, −2.65883473144462392459839017043, −0.74745755211716692610772701923,
0.22209249059355366098538954002, 1.70451132432863807924302709996, 3.03061619013472631914891315881, 3.53437761784028678766360960763, 4.60965223868030921285037204972, 5.76904760750194826630963391705, 6.09077616887823631630613589797, 7.20975432361341106679112197268, 7.969117860047172467668598216196, 8.438721282166891288916406118309
