L(s) = 1 | + (−1.58 − 2.73i)5-s + (−2.5 + 0.866i)7-s + (−3.16 + 5.47i)11-s + (1.58 + 2.73i)17-s + (3.5 − 6.06i)19-s + (1.58 + 2.73i)23-s + (−2.5 + 4.33i)25-s + (−1.58 − 2.73i)29-s + 3·31-s + (6.32 + 5.47i)35-s + (2 − 3.46i)37-s + (−4.74 + 8.21i)41-s + (−2.5 − 4.33i)43-s + 9.48·47-s + (5.5 − 4.33i)49-s + ⋯ |
L(s) = 1 | + (−0.707 − 1.22i)5-s + (−0.944 + 0.327i)7-s + (−0.953 + 1.65i)11-s + (0.383 + 0.664i)17-s + (0.802 − 1.39i)19-s + (0.329 + 0.571i)23-s + (−0.5 + 0.866i)25-s + (−0.293 − 0.508i)29-s + 0.538·31-s + (1.06 + 0.925i)35-s + (0.328 − 0.569i)37-s + (−0.740 + 1.28i)41-s + (−0.381 − 0.660i)43-s + 1.38·47-s + (0.785 − 0.618i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.086340084\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.086340084\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 5 | \( 1 + (1.58 + 2.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.16 - 5.47i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.58 - 2.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.58 - 2.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.58 + 2.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.74 - 8.21i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.48T + 47T^{2} \) |
| 53 | \( 1 + (4.74 + 8.21i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + (3.16 + 5.47i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.74 + 8.21i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.997179886043023283022814575592, −8.201883507336107293432992799297, −7.45700112215667012852584633455, −6.81944722741091340699175551493, −5.59169032291273526482306671408, −4.95437886157432062570921442606, −4.24551427086481144630427576594, −3.18327942597489053528272552807, −2.08871733125093530880032967807, −0.61533604339714317972015828651,
0.70034640496425418304746101864, 2.72927980428415247387415035013, 3.24292490018944271136677113759, 3.84354619329380550961766399076, 5.29488817595508450836703310802, 6.03739995044504978263722118305, 6.79949752705532735666226304853, 7.55544804083418862901601742556, 8.110468827751825299562578507380, 9.060261838918193063759194130196