L(s) = 1 | − 30.1i·3-s + 55.9·5-s + 194. i·7-s − 667.·9-s − 1.68e3i·15-s + 5.86e3·21-s + 5.06e3i·23-s + 3.12e3·25-s + 1.28e4i·27-s − 1.68e3·29-s + 1.08e4i·35-s + 2.10e4·41-s + 1.01e4i·43-s − 3.73e4·45-s + 3.21e3i·47-s + ⋯ |
L(s) = 1 | − 1.93i·3-s + 0.999·5-s + 1.49i·7-s − 2.74·9-s − 1.93i·15-s + 2.89·21-s + 1.99i·23-s + 25-s + 3.38i·27-s − 0.372·29-s + 1.49i·35-s + 1.95·41-s + 0.837i·43-s − 2.74·45-s + 0.212i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.972645972\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.972645972\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 55.9T \) |
good | 3 | \( 1 + 30.1iT - 243T^{2} \) |
| 7 | \( 1 - 194. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 1.61e5T^{2} \) |
| 13 | \( 1 - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.47e6T^{2} \) |
| 23 | \( 1 - 5.06e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.68e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.93e7T^{2} \) |
| 41 | \( 1 - 2.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.01e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.21e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 4.18e8T^{2} \) |
| 59 | \( 1 + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.22e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.35e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.16e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.49e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.58e9T^{2} \) |
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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
−11.23840591002921423643807703180, −9.535184176654856440977763745549, −8.826857318040689824736983022122, −7.85771784620067031442391585798, −6.87776193681051029686958129355, −5.80513218378867535521654327306, −5.56089941471565918349886250617, −2.88549434951700315025926441153, −2.10207849389218407270804125482, −1.20492241706955961691313177704,
0.53770298992058534174940357216, 2.59665113258588841773122841067, 3.87911010103700635423855442789, 4.58326230182219684256444453552, 5.60110135330737652909960761924, 6.72737497975470380956938186711, 8.279552753779466766791288894968, 9.271522827850503363751293984789, 9.994548441109771740520175238692, 10.61767889651437474590897271727