| L(s) = 1 | − 0.414i·2-s + 1.82·4-s + 2i·7-s − 1.58i·8-s − 11-s − 6.82i·13-s + 0.828·14-s + 3·16-s + 1.17i·17-s + 0.414i·22-s − 2.82i·23-s − 2.82·26-s + 3.65i·28-s + 7.65·29-s − 4.41i·32-s + ⋯ |
| L(s) = 1 | − 0.292i·2-s + 0.914·4-s + 0.755i·7-s − 0.560i·8-s − 0.301·11-s − 1.89i·13-s + 0.221·14-s + 0.750·16-s + 0.284i·17-s + 0.0883i·22-s − 0.589i·23-s − 0.554·26-s + 0.691i·28-s + 1.42·29-s − 0.780i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.219064965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.219064965\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 13 | \( 1 + 6.82iT - 13T^{2} \) |
| 17 | \( 1 - 1.17iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 3.65iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 0.343iT - 53T^{2} \) |
| 59 | \( 1 + 9.65T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 4.48iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 6.82iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 9.31T + 89T^{2} \) |
| 97 | \( 1 - 7.65iT - 97T^{2} \) |
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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
−8.593256392646559901818866940667, −8.128427551566425840880077843861, −7.27966064917809597414446013002, −6.42157375500980571693457986590, −5.69516050778051913047201212083, −5.03816391174482433433435800034, −3.64625957398858070236307501710, −2.85277762450337239895914591725, −2.17230378619256087587225470884, −0.76304003956806212336008103255,
1.28331792487326790948264980977, 2.26788405973561448328255368373, 3.31768717707934086109666035062, 4.34485160229358618802050751053, 5.11859164278606947358614517544, 6.30002296784641048810143140823, 6.72796732546991128303889864118, 7.40096254286805313825791904710, 8.136315308133945051733557045401, 9.006715202777046397104089970207
