L(s) = 1 | + (−1.58 + 1.58i)2-s − 3.00i·4-s + (−2 − i)5-s + (2.58 + 0.581i)7-s + (1.58 + 1.58i)8-s + (4.74 − 1.58i)10-s + 3.16·11-s + (−3.16 + 3.16i)13-s + (−5 + 3.16i)14-s + 0.999·16-s + (5 + 5i)17-s − 3.16·19-s + (−3.00 + 6.00i)20-s + (−5.00 + 5.00i)22-s + (3.16 + 3.16i)23-s + ⋯ |
L(s) = 1 | + (−1.11 + 1.11i)2-s − 1.50i·4-s + (−0.894 − 0.447i)5-s + (0.975 + 0.219i)7-s + (0.559 + 0.559i)8-s + (1.50 − 0.500i)10-s + 0.953·11-s + (−0.877 + 0.877i)13-s + (−1.33 + 0.845i)14-s + 0.249·16-s + (1.21 + 1.21i)17-s − 0.725·19-s + (−0.670 + 1.34i)20-s + (−1.06 + 1.06i)22-s + (0.659 + 0.659i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.385578 + 0.540850i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.385578 + 0.540850i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 + (-2.58 - 0.581i)T \) |
good | 2 | \( 1 + (1.58 - 1.58i)T - 2iT^{2} \) |
| 11 | \( 1 - 3.16T + 11T^{2} \) |
| 13 | \( 1 + (3.16 - 3.16i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5 - 5i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.16T + 19T^{2} \) |
| 23 | \( 1 + (-3.16 - 3.16i)T + 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 3.16iT - 31T^{2} \) |
| 37 | \( 1 + (3 - 3i)T - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-6 - 6i)T + 43iT^{2} \) |
| 47 | \( 1 + 47iT^{2} \) |
| 53 | \( 1 + (3.16 + 3.16i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12.6iT - 61T^{2} \) |
| 67 | \( 1 + (-8 + 8i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.48T + 71T^{2} \) |
| 73 | \( 1 + (-9.48 + 9.48i)T - 73iT^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + (10 - 10i)T - 83iT^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + (3.16 + 3.16i)T + 97iT^{2} \) |
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
−11.85769444411272896997630198721, −10.90595190382898474224760110741, −9.651614571102862027392937864585, −8.877243308460527328936255216283, −8.090449833856409722741730905586, −7.45166518771918788046864920065, −6.40185125735926098774992360905, −5.16045205811813834294947732006, −3.95620997818703823167092281589, −1.38863580016605086462133556632,
0.813889367477609791797144882235, 2.54096479173390942167552602862, 3.71449594595991661481447573639, 5.10358975583938167746688007002, 7.05005342763896151610138316944, 7.83366746811300733846419968206, 8.624414539095412286580194563500, 9.663768395935485095267127476523, 10.61347109285983570090878254492, 11.17488243602496365956114594996