L(s) = 1 | − 268. i·3-s − 637. i·7-s − 5.24e4·9-s + 4.90e4·11-s − 7.27e4i·13-s + 6.73e4i·17-s + 3.41e5·19-s − 1.71e5·21-s − 1.34e5i·23-s + 8.81e6i·27-s − 4.45e6·29-s − 4.56e5·31-s − 1.31e7i·33-s + 1.30e7i·37-s − 1.95e7·39-s + ⋯ |
L(s) = 1 | − 1.91i·3-s − 0.100i·7-s − 2.66·9-s + 1.00·11-s − 0.706i·13-s + 0.195i·17-s + 0.600·19-s − 0.192·21-s − 0.100i·23-s + 3.19i·27-s − 1.17·29-s − 0.0887·31-s − 1.93i·33-s + 1.14i·37-s − 1.35·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.507445123\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.507445123\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 268. iT - 1.96e4T^{2} \) |
| 7 | \( 1 + 637. iT - 4.03e7T^{2} \) |
| 11 | \( 1 - 4.90e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.27e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 6.73e4iT - 1.18e11T^{2} \) |
| 19 | \( 1 - 3.41e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.34e5iT - 1.80e12T^{2} \) |
| 29 | \( 1 + 4.45e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.56e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.30e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 2.56e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.42e6iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 3.39e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 - 8.42e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 7.46e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.78e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.94e7iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 2.07e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.02e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 - 3.72e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.50e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 5.82e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.85e8iT - 7.60e17T^{2} \) |
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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
−9.479106613956452758092287958374, −8.490670968702649639534258527202, −7.73837263274847568635099043319, −6.94314978380085602795139991253, −6.20537245415881483520461039812, −5.30367736508203334278249856881, −3.59781302693165884322551604036, −2.54902659804746315498872973674, −1.46571376687663524340359609382, −0.857807587561656437438427072657,
0.32263862312116705775903853131, 2.03464283437350016911880813373, 3.44441584508934158966610573332, 3.94310635609647069876504552275, 4.97020286264555513495563663171, 5.74066860903933559754492245271, 6.95088505384704750586739579898, 8.418111301598675490343687206134, 9.227433952898093371891930564424, 9.659480280245259221561462774740