| L(s) = 1 | + i·2-s + 4-s + 2i·7-s + 3i·8-s + 3·9-s − 6·11-s + 2i·13-s − 2·14-s − 16-s + 2i·17-s + 3i·18-s + 2·19-s − 6i·22-s + 2i·23-s − 2·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.5·4-s + 0.755i·7-s + 1.06i·8-s + 9-s − 1.80·11-s + 0.554i·13-s − 0.534·14-s − 0.250·16-s + 0.485i·17-s + 0.707i·18-s + 0.458·19-s − 1.27i·22-s + 0.417i·23-s − 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.880403 + 1.42452i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.880403 + 1.42452i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−10.67312196610127919462588598658, −9.833167495559740938355149837873, −8.790289877698730065584705724601, −7.77598778692835818643125792560, −7.37827757031045747278894894102, −6.22198069079057810508297735477, −5.49969097173598607800637660478, −4.56063402080358753068589588525, −2.92347578162155466803677423313, −1.91263034095113813180296094491,
0.862835805039292746039049720786, 2.34101009603683483821703918774, 3.29871767503832252663994663052, 4.46914927729868417741466153421, 5.51051609302843871070980223670, 6.87980060839074380493989575229, 7.40411307127876632890816351909, 8.276166332733417785109124347038, 9.776237915665878080063535621343, 10.34072067206267059604203245946
