L(s) = 1 | + (−0.581 − 1.28i)2-s + (−1.32 + 1.50i)4-s + (−1.41 + 1.41i)7-s + (2.70 + 0.832i)8-s − 5.29i·11-s + (−3.74 + 3.74i)13-s + (2.64 + 1.00i)14-s + (−0.5 − 3.96i)16-s + 5.29·19-s + (−6.82 + 3.07i)22-s + (2.82 + 2.82i)23-s + (7 + 2.64i)26-s + (−0.250 − 3.99i)28-s + 8i·29-s + 5.29i·31-s + (−4.82 + 2.95i)32-s + ⋯ |
L(s) = 1 | + (−0.411 − 0.911i)2-s + (−0.661 + 0.750i)4-s + (−0.534 + 0.534i)7-s + (0.955 + 0.294i)8-s − 1.59i·11-s + (−1.03 + 1.03i)13-s + (0.707 + 0.267i)14-s + (−0.125 − 0.992i)16-s + 1.21·19-s + (−1.45 + 0.656i)22-s + (0.589 + 0.589i)23-s + (1.37 + 0.518i)26-s + (−0.0473 − 0.754i)28-s + 1.48i·29-s + 0.950i·31-s + (−0.852 + 0.522i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.888148 + 0.131727i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.888148 + 0.131727i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.581 + 1.28i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.41 - 1.41i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.29iT - 11T^{2} \) |
| 13 | \( 1 + (3.74 - 3.74i)T - 13iT^{2} \) |
| 17 | \( 1 + 17iT^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 - 8iT - 29T^{2} \) |
| 31 | \( 1 - 5.29iT - 31T^{2} \) |
| 37 | \( 1 + (-3.74 - 3.74i)T + 37iT^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + (-5.65 - 5.65i)T + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (7.48 - 7.48i)T - 53iT^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + (-8.48 + 8.48i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (7.48 - 7.48i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + (-7.48 - 7.48i)T + 97iT^{2} \) |
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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
−10.08015328720688117186897049368, −9.245275660529749377108838447126, −8.886862393672558845700980614541, −7.79069954927742579480650081756, −6.88447049250603285156975299265, −5.66125379706922023479424868767, −4.71546335681051712314083551037, −3.36332403471069279487046346374, −2.78481231398033204153518657773, −1.21798168931629959100481498371,
0.56629082300339214168471968151, 2.39754540234644671414712114216, 4.01621731883193360265056280176, 4.92317233968516039122818487183, 5.78093186036446749339476805534, 6.96579907188905254603654643390, 7.40034517606791286078826526603, 8.130202352709596776028906514650, 9.513866527121474424847246206455, 9.776263341647845908796822167175