| L(s) = 1 | − i·5-s + 2·7-s + 5.46i·11-s − 5.46i·13-s − 3.46·17-s − 6.92i·19-s − 4·23-s − 25-s − 8.92i·29-s + 0.535·31-s − 2i·35-s + 9.46i·37-s − 4.92·41-s − 6.92i·43-s + 4·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 0.755·7-s + 1.64i·11-s − 1.51i·13-s − 0.840·17-s − 1.58i·19-s − 0.834·23-s − 0.200·25-s − 1.65i·29-s + 0.0962·31-s − 0.338i·35-s + 1.55i·37-s − 0.769·41-s − 1.05i·43-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.367144244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367144244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 5.46iT - 11T^{2} \) |
| 13 | \( 1 + 5.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 8.92iT - 29T^{2} \) |
| 31 | \( 1 - 0.535T + 31T^{2} \) |
| 37 | \( 1 - 9.46iT - 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 + 6.92iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 0.928iT - 53T^{2} \) |
| 59 | \( 1 + 9.46iT - 59T^{2} \) |
| 61 | \( 1 + 4iT - 61T^{2} \) |
| 67 | \( 1 + 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.288781922404387532779043572202, −8.013252399336281385234741809876, −7.08565505018109854100365568317, −6.34914627987320559055174455027, −5.12441794841040612610434075050, −4.84156584452311496321614099875, −3.95740109161424600958878042639, −2.60850547370271701063099286094, −1.85167489004462808845205779508, −0.42789353486627339010793441804,
1.38168833860626877442679926486, 2.29497942907649975715942238321, 3.53821733549615832378069818729, 4.11505483205392322379107379612, 5.20074257672030936844480562535, 6.02575461302860040565243337818, 6.62810097566329494783720872900, 7.53800762996341175763750344244, 8.337433759116609653869164328576, 8.838596671816668850192981073518
