L(s) = 1 | − 0.874i·3-s + 2.23·5-s + 2.82i·7-s + 2.23·9-s + 0.763·11-s − 5.45i·13-s − 1.95i·15-s + 7.40i·17-s − 19-s + 2.47·21-s + 1.08i·23-s + 5.00·25-s − 4.57i·27-s + 4.47·29-s + 4·31-s + ⋯ |
L(s) = 1 | − 0.504i·3-s + 0.999·5-s + 1.06i·7-s + 0.745·9-s + 0.230·11-s − 1.51i·13-s − 0.504i·15-s + 1.79i·17-s − 0.229·19-s + 0.539·21-s + 0.225i·23-s + 1.00·25-s − 0.880i·27-s + 0.830·29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.256127227\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.256127227\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 0.874iT - 3T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 + 5.45iT - 13T^{2} \) |
| 17 | \( 1 - 7.40iT - 17T^{2} \) |
| 23 | \( 1 - 1.08iT - 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2.62iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 - 8.48iT - 47T^{2} \) |
| 53 | \( 1 - 2.62iT - 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 - 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 5.24iT - 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 13.7iT - 83T^{2} \) |
| 89 | \( 1 + 2.94T + 89T^{2} \) |
| 97 | \( 1 + 13.9iT - 97T^{2} \) |
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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
−9.505311258893448360383406267947, −8.528384915927602925103755657517, −8.072531279329500846321863866102, −6.89199686909356662442319023055, −6.08678966076541757880115864477, −5.63888996969578368688672453369, −4.55380762098603427637189860559, −3.20096910062222297460504201692, −2.19174949254656807452003272147, −1.27329487305174095836253820266,
1.08853593004070106684044245866, 2.26374560378313040173124509091, 3.57236117439916879968614843310, 4.60092908589301462952784790211, 4.98178539175047273138265417363, 6.59405542835070585831169088581, 6.73338720964366953546835899141, 7.76494788730919408381344375565, 9.077632763134642413390577396688, 9.471766248116396315255224588331