L(s) = 1 | + (0.994 − 2.00i)5-s + (−2.91 − 2.91i)7-s + 4.72i·11-s + (0.839 − 0.839i)13-s + (5.22 − 5.22i)17-s − 3.12i·19-s + (0.707 + 0.707i)23-s + (−3.02 − 3.98i)25-s − 4.03·29-s − 6.68·31-s + (−8.74 + 2.94i)35-s + (2.81 + 2.81i)37-s − 9.02i·41-s + (1.64 − 1.64i)43-s + (1.14 − 1.14i)47-s + ⋯ |
L(s) = 1 | + (0.444 − 0.895i)5-s + (−1.10 − 1.10i)7-s + 1.42i·11-s + (0.232 − 0.232i)13-s + (1.26 − 1.26i)17-s − 0.716i·19-s + (0.147 + 0.147i)23-s + (−0.604 − 0.796i)25-s − 0.748·29-s − 1.20·31-s + (−1.47 + 0.497i)35-s + (0.463 + 0.463i)37-s − 1.40i·41-s + (0.251 − 0.251i)43-s + (0.167 − 0.167i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.060204808\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.060204808\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.994 + 2.00i)T \) |
| 23 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + (2.91 + 2.91i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.72iT - 11T^{2} \) |
| 13 | \( 1 + (-0.839 + 0.839i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5.22 + 5.22i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.12iT - 19T^{2} \) |
| 29 | \( 1 + 4.03T + 29T^{2} \) |
| 31 | \( 1 + 6.68T + 31T^{2} \) |
| 37 | \( 1 + (-2.81 - 2.81i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.02iT - 41T^{2} \) |
| 43 | \( 1 + (-1.64 + 1.64i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.14 + 1.14i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.243 - 0.243i)T + 53iT^{2} \) |
| 59 | \( 1 + 3.07T + 59T^{2} \) |
| 61 | \( 1 - 8.53T + 61T^{2} \) |
| 67 | \( 1 + (4.44 + 4.44i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.08iT - 71T^{2} \) |
| 73 | \( 1 + (1.30 - 1.30i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.837iT - 79T^{2} \) |
| 83 | \( 1 + (1.87 + 1.87i)T + 83iT^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 + (-5.02 - 5.02i)T + 97iT^{2} \) |
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
−7.927692441358795917543656620472, −7.21590284233655294734142280979, −6.88562140307107889077613798833, −5.71304059262355161508002844626, −5.14152703466414615351917671230, −4.27048025777769812882033001818, −3.54784132178341847652182487879, −2.48778806795062856537165217949, −1.30772291454745156138007206694, −0.30630680042292876345817952939,
1.48249218256556669995824349486, 2.60928499859133963208067607772, 3.33301130298205334386027624693, 3.80475173164974252803253149606, 5.49604348986692910971420394591, 5.96200913104638396760720045690, 6.20267763021873461183396494259, 7.23042645975770292715245770460, 8.114521810180734478209324174110, 8.748854500506665287493720324524