L(s) = 1 | − 3-s + 5-s + 9-s + 3.46i·11-s − 15-s + (4 + 1.73i)19-s − 3.46i·23-s + 25-s − 27-s − 6.92i·29-s + 2·31-s − 3.46i·33-s − 6.92i·37-s + 3.46i·41-s + 10.3i·43-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.333·9-s + 1.04i·11-s − 0.258·15-s + (0.917 + 0.397i)19-s − 0.722i·23-s + 0.200·25-s − 0.192·27-s − 1.28i·29-s + 0.359·31-s − 0.603i·33-s − 1.13i·37-s + 0.541i·41-s + 1.58i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 - 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.688062213\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.688062213\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 + 10.3iT - 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−8.341147327758749115538787649213, −7.54125323313774272517193289300, −6.96463524476219341932812577747, −6.10482747075722118112335732785, −5.58473993323060728155509761557, −4.65715494166567413702172273426, −4.11825962885721029810828318202, −2.86205684627799366378172938684, −1.98793297988875652447190692327, −0.900495170910319867699830737912,
0.65940472143622125388689901843, 1.65642787595182014946189242962, 2.90046453652610337549017634835, 3.61580191497167478409409978411, 4.68750815163490212377674520441, 5.55881970454292824947434639817, 5.78452088221382407678548648991, 6.92534094657705784984949006646, 7.24809641918808843436232034512, 8.437775599658664199421713722645