L(s) = 1 | + (−1.53 − 0.5i)2-s + (−0.587 + 0.809i)3-s + (0.5 + 0.363i)4-s + (1.30 − 0.951i)6-s + 0.618i·7-s + (1.31 + 1.80i)8-s + (0.618 + 1.90i)9-s + (−1.61 + 4.97i)11-s + (−0.587 + 0.190i)12-s + (1.76 − 0.572i)13-s + (0.309 − 0.951i)14-s + (−1.50 − 4.61i)16-s + (3.07 + 4.23i)17-s − 3.23i·18-s + (0.690 − 0.502i)19-s + ⋯ |
L(s) = 1 | + (−1.08 − 0.353i)2-s + (−0.339 + 0.467i)3-s + (0.250 + 0.181i)4-s + (0.534 − 0.388i)6-s + 0.233i·7-s + (0.464 + 0.639i)8-s + (0.206 + 0.634i)9-s + (−0.487 + 1.50i)11-s + (−0.169 + 0.0551i)12-s + (0.489 − 0.158i)13-s + (0.0825 − 0.254i)14-s + (−0.375 − 1.15i)16-s + (0.746 + 1.02i)17-s − 0.762i·18-s + (0.158 − 0.115i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.421592 + 0.264041i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.421592 + 0.264041i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (1.53 + 0.5i)T + (1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (0.587 - 0.809i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 - 0.618iT - 7T^{2} \) |
| 11 | \( 1 + (1.61 - 4.97i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.76 + 0.572i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-3.07 - 4.23i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.690 + 0.502i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (3.57 + 1.16i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (2.92 + 2.12i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.42 + 1.76i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.224 + 0.0729i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.236 + 0.726i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 4.85iT - 43T^{2} \) |
| 47 | \( 1 + (-0.363 + 0.5i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.04 + 2.80i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.35 - 10.3i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.69 + 8.28i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (2.80 + 3.85i)T + (-20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-5.35 - 3.88i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-8.55 - 2.78i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (6.54 + 4.75i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (3.66 + 5.04i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.76 + 8.50i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (2.26 - 3.11i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−13.46150816842799056782442531005, −12.32403665881400707431415693133, −11.13224830495819489446084991521, −10.20760026087833821465771107326, −9.752493802480971913766699199979, −8.351649119374772113152207343409, −7.49592973576757152531307133444, −5.62349778898807181124067738562, −4.40696189794753581492843109110, −2.00941807045984137695360907115,
0.835682917278180444937163876946, 3.62812310243019469106802979994, 5.69746366203112538529994426349, 6.88441802301984544498556194121, 7.88236412906786309843159776419, 8.874270316622056629678748612294, 9.893628305948710679568794857429, 11.00466496761026649724976613397, 12.07060420389777768739380127348, 13.27783753121658280529549743903