L(s) = 1 | + 19.8i·3-s + (−45 + 33.1i)5-s + 59.6i·7-s − 153·9-s − 252·11-s − 119. i·13-s + (−660 − 895. i)15-s − 689. i·17-s + 220·19-s − 1.18e3·21-s − 2.43e3i·23-s + (924. − 2.98e3i)25-s + 1.79e3i·27-s − 6.93e3·29-s − 6.75e3·31-s + ⋯ |
L(s) = 1 | + 1.27i·3-s + (−0.804 + 0.593i)5-s + 0.460i·7-s − 0.629·9-s − 0.627·11-s − 0.195i·13-s + (−0.757 − 1.02i)15-s − 0.578i·17-s + 0.139·19-s − 0.587·21-s − 0.959i·23-s + (0.295 − 0.955i)25-s + 0.472i·27-s − 1.53·29-s − 1.26·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.183205 - 0.557366i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.183205 - 0.557366i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (45 - 33.1i)T \) |
good | 3 | \( 1 - 19.8iT - 243T^{2} \) |
| 7 | \( 1 - 59.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 252T + 1.61e5T^{2} \) |
| 13 | \( 1 + 119. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 689. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 220T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.43e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 6.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 198T + 1.15e8T^{2} \) |
| 43 | \( 1 - 417. iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.05e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 5.82e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.36e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.09e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 5.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.18e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.99e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.01e5iT - 8.58e9T^{2} \) |
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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
−14.49752594411039763082254060543, −12.90983674890392720471875134896, −11.58005729088650790866170375174, −10.73355844752777835638069157217, −9.770690845478940023450445049360, −8.540671969408024928511524549821, −7.20400959014858528732836540970, −5.43403235251834930132038685870, −4.16887342677639859435482830729, −2.90215463686548024180531290082,
0.24545829951886585968100040249, 1.72972365910079842681397333888, 3.83913336687543553481393853115, 5.58315848127763899319032472826, 7.25338803591288837699092310252, 7.75938724834058794437686308689, 9.095376398425732577586925992651, 10.81171708991485132610084415289, 11.90777150247344623859242650377, 12.85937452426659599286633648871