| L(s) = 1 | − 5.19i·3-s − 30.1·5-s − 52.3i·7-s − 27·9-s − 90.0i·11-s + 60.3·13-s + 156. i·15-s − 338·17-s + 6.92i·19-s − 271.·21-s − 732. i·23-s + 287·25-s + 140. i·27-s − 1.29e3·29-s − 1.30e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.20·5-s − 1.06i·7-s − 0.333·9-s − 0.744i·11-s + 0.357·13-s + 0.697i·15-s − 1.16·17-s + 0.0191i·19-s − 0.616·21-s − 1.38i·23-s + 0.459·25-s + 0.192i·27-s − 1.54·29-s − 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2669631042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2669631042\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19iT \) |
| good | 5 | \( 1 + 30.1T + 625T^{2} \) |
| 7 | \( 1 + 52.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 90.0iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 60.3T + 2.85e4T^{2} \) |
| 17 | \( 1 + 338T + 8.35e4T^{2} \) |
| 19 | \( 1 - 6.92iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 732. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.29e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.30e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 241.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 578T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.02e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.19e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.44e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.19e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 6.40e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 8.26e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.28e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.73e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.12e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.31e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 910T + 6.27e7T^{2} \) |
| 97 | \( 1 - 5.42e3T + 8.85e7T^{2} \) |
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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
−8.748097595217944626326222061950, −8.245935865898234278684894392081, −7.28594240755213529955973416292, −6.77912655436952432939626982950, −5.60508542020240886304480245661, −4.17365224630341745785129908454, −3.75219755275025533851424724172, −2.30522793020256424482689170716, −0.74674842724383642739842797655, −0.087077796101569343193217629408,
1.81239282249067879588535914009, 3.11748534027405823468767390212, 4.02769793535978531783794379293, 4.90793121911743820416781898895, 5.85842700104597520386477092950, 7.04634910667388870904596023769, 7.87002023520587737221787100057, 8.834837528484841756190302434519, 9.312293865683730209917234612841, 10.43491508590651917624744075486
