L(s) = 1 | − 1.59·5-s − 1.16i·7-s − 1.98i·11-s + 0.164i·13-s + 3.57i·17-s + 3.16·19-s − 1.21·23-s − 2.44·25-s − 4.98·29-s + 6.64i·31-s + 1.85i·35-s − 1.10i·37-s + 2.02i·41-s + 8.34·43-s + 2.21·47-s + ⋯ |
L(s) = 1 | − 0.714·5-s − 0.440i·7-s − 0.597i·11-s + 0.0455i·13-s + 0.867i·17-s + 0.725·19-s − 0.253·23-s − 0.489·25-s − 0.925·29-s + 1.19i·31-s + 0.314i·35-s − 0.181i·37-s + 0.316i·41-s + 1.27·43-s + 0.323·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.448042824\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.448042824\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.59T + 5T^{2} \) |
| 7 | \( 1 + 1.16iT - 7T^{2} \) |
| 11 | \( 1 + 1.98iT - 11T^{2} \) |
| 13 | \( 1 - 0.164iT - 13T^{2} \) |
| 17 | \( 1 - 3.57iT - 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 + 1.21T + 23T^{2} \) |
| 29 | \( 1 + 4.98T + 29T^{2} \) |
| 31 | \( 1 - 6.64iT - 31T^{2} \) |
| 37 | \( 1 + 1.10iT - 37T^{2} \) |
| 41 | \( 1 - 2.02iT - 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 47 | \( 1 - 2.21T + 47T^{2} \) |
| 53 | \( 1 - 9.65T + 53T^{2} \) |
| 59 | \( 1 - 0.888iT - 59T^{2} \) |
| 61 | \( 1 + 9.74iT - 61T^{2} \) |
| 67 | \( 1 - 3.98T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 0.879T + 73T^{2} \) |
| 79 | \( 1 - 2.91iT - 79T^{2} \) |
| 83 | \( 1 + 13.9iT - 83T^{2} \) |
| 89 | \( 1 - 8.41iT - 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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
−8.084629164823716879232384042784, −7.52952404749229003224248257508, −6.88799873135203886646418898849, −5.94673790005817080228927603113, −5.36807367771880115381676291173, −4.25877843033966249714750674043, −3.78066425187015391495389624234, −2.98384745878855153293280679187, −1.75763749613619184942337450237, −0.62891944940105190531701394501,
0.67097078866003187244621622611, 2.04989643545942797363863910049, 2.84038073716652330076265628574, 3.88256665634514453654019322564, 4.40986124093346919315866715718, 5.45934242388039227476384406183, 5.89032947145676673736336263925, 7.15176017056407610294620286954, 7.39118813418436855587219489611, 8.141564447988226903378156492962