L(s) = 1 | + (0.809 − 0.587i)2-s + (0.618 + 1.90i)3-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + (1.61 + 1.17i)6-s + (0.618 − 1.90i)7-s + (0.927 + 2.85i)8-s + (−0.809 + 0.587i)9-s − 10-s − 1.99·12-s + (0.809 − 0.587i)13-s + (−0.618 − 1.90i)14-s + (0.618 − 1.90i)15-s + (0.809 + 0.587i)16-s + (−4.04 − 2.93i)17-s + (−0.309 + 0.951i)18-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (0.356 + 1.09i)3-s + (−0.154 + 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.660 + 0.479i)6-s + (0.233 − 0.718i)7-s + (0.327 + 1.00i)8-s + (−0.269 + 0.195i)9-s − 0.316·10-s − 0.577·12-s + (0.224 − 0.163i)13-s + (−0.165 − 0.508i)14-s + (0.159 − 0.491i)15-s + (0.202 + 0.146i)16-s + (−0.981 − 0.712i)17-s + (−0.0728 + 0.224i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.36911 + 0.376555i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.36911 + 0.376555i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.618 - 1.90i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (0.809 + 0.587i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.618 + 1.90i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-0.809 + 0.587i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (4.04 + 2.93i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.85 + 5.70i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + (2.78 - 8.55i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.61 + 1.17i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.927 - 2.85i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.54 - 4.75i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (-0.618 - 1.90i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (7.28 - 5.29i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.47 + 7.60i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.85 - 3.52i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + (9.70 + 7.05i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.618 + 1.90i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (8.09 - 5.87i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.85 - 3.52i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + (-10.5 + 7.64i)T + (29.9 - 92.2i)T^{2} \) |
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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
−13.51354300739221769935451639042, −12.65465774129199013639471913502, −11.34616612103008001366064837771, −10.70489484640390481652921542669, −9.287048105505832980344628472181, −8.436618857388649544125903002040, −7.07909311579771284569498995016, −4.82177005491798371864729318047, −4.27241533587434742940651276809, −3.01375699281306565671805504984,
1.92659410493947734409567211885, 4.06639688122683709048418171842, 5.70061139574055580956689523721, 6.64579030538668847188487540763, 7.75384528321022666444419855021, 8.854029960385424933647284807661, 10.29843181471543889399007878186, 11.58306610875824675153532381267, 12.72528376820201862976593573944, 13.36698931711887760339571072178