L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (−1.61 − 1.17i)5-s + (−0.809 − 0.587i)6-s + (1.23 − 3.80i)7-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s − 2·10-s − 12-s + (−4.85 + 3.52i)13-s + (−1.23 − 3.80i)14-s + (−0.618 + 1.90i)15-s + (−0.809 − 0.587i)16-s + (1.61 + 1.17i)17-s + (−0.309 + 0.951i)18-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (−0.723 − 0.525i)5-s + (−0.330 − 0.239i)6-s + (0.467 − 1.43i)7-s + (−0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s − 0.632·10-s − 0.288·12-s + (−1.34 + 0.978i)13-s + (−0.330 − 1.01i)14-s + (−0.159 + 0.491i)15-s + (−0.202 − 0.146i)16-s + (0.392 + 0.285i)17-s + (−0.0728 + 0.224i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0290862 - 1.32274i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0290862 - 1.32274i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + (1.61 + 1.17i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-1.23 + 3.80i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (4.85 - 3.52i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.61 - 1.17i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.23 + 3.80i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + (1.85 - 5.70i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.85 + 5.70i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.85 - 5.70i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (3.70 + 11.4i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.61 - 1.17i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.70 + 11.4i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (11.3 + 8.22i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + (-9.70 - 7.05i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.85 + 5.70i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (3.23 - 2.35i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-3.23 - 2.35i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + (-11.3 + 8.22i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.19889737649590945662677215915, −9.186652324706758979024254209223, −8.033526945742341211337129944014, −7.23661739472069406136959396019, −6.65920556511018437009538103593, −5.01889531368818392887774126087, −4.56324840078504473949385846273, −3.50151448774409285127038807124, −1.93092438828697569362380574756, −0.56856086255849240854545267999,
2.50810510863717023992916304830, 3.35791088458841686369034950485, 4.64742823935695721709111380936, 5.38038101860193600850038224307, 6.15633372611959802307947375798, 7.48621696074675215363544613113, 8.003899982460967928302767493930, 9.053206303408370327663145022446, 9.986518659634865056218497730405, 10.97706996111485679570045588550