| L(s) = 1 | + 2i·7-s + 3·9-s + 11-s − 8·19-s − 8i·23-s − 10·29-s − 8·31-s − 10i·37-s − 2·41-s − 6i·43-s + 8i·47-s + 3·49-s − 14i·53-s − 4·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.755i·7-s + 9-s + 0.301·11-s − 1.83·19-s − 1.66i·23-s − 1.85·29-s − 1.43·31-s − 1.64i·37-s − 0.312·41-s − 0.914i·43-s + 1.16i·47-s + 0.428·49-s − 1.92i·53-s − 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8620551940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8620551940\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 14iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 10iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.233534680830904802576332703701, −7.28353569542483018708836282952, −6.73015654793833406618220027928, −5.93741964595754633704130381082, −5.20787547662399056070055629642, −4.20353733430988535345696739498, −3.76062417333265384353872895755, −2.32129383064925947799850511398, −1.86237394473426923529474242763, −0.22942666032940754763124551861,
1.32205053360290164875469187472, 2.03512810024324411219103687305, 3.49575347005544356926014876528, 3.99192640734493392743961311446, 4.74486459160823184004287558951, 5.70844906895903716326390942239, 6.50012029258434021074980432500, 7.29753960001246101262495865526, 7.60653758125453381460823461758, 8.643537153037795994690228853748
