L(s) = 1 | + 4i·2-s − 16·4-s − 98i·7-s − 64i·8-s − 354·11-s + 404i·13-s + 392·14-s + 256·16-s + 654i·17-s − 1.79e3·19-s − 1.41e3i·22-s + 1.08e3i·23-s − 1.61e3·26-s + 1.56e3i·28-s − 5.75e3·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.755i·7-s − 0.353i·8-s − 0.882·11-s + 0.663i·13-s + 0.534·14-s + 0.250·16-s + 0.548i·17-s − 1.14·19-s − 0.623i·22-s + 0.425i·23-s − 0.468·26-s + 0.377i·28-s − 1.27·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.487134853\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.487134853\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 98iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 354T + 1.61e5T^{2} \) |
| 13 | \( 1 - 404iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 654iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.08e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 5.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.01e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.55e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.29e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.96e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 5.40e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 8.21e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.95e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 962T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.79e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.61e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.56e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 2.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.34e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 7.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.28e5iT - 8.58e9T^{2} \) |
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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
−10.34219346670654745362660129872, −9.364117298900427358986604649774, −8.353678078345244855121736809398, −7.58171582189101263195110343002, −6.69354897610083707626986067754, −5.75348664454641273635208613074, −4.59940060664626995630977527685, −3.75244587907017742849056603563, −2.14348545924013479149918571766, −0.56070629357527069293556852313,
0.67309541194531818486015002203, 2.24496994235425324259953094475, 2.95222965365990008370619645064, 4.37541871248185439060241844236, 5.33840409146129457554554047888, 6.30745054784354111580924889557, 7.75318129817252636597864626224, 8.486584387266337560833474699133, 9.442535322036218021477214214738, 10.32201929136535336802421754363