L(s) = 1 | + (1.71 − 0.272i)3-s + (−0.119 − 0.207i)5-s + (−0.5 + 0.866i)7-s + (2.85 − 0.931i)9-s + (−2.56 + 4.43i)11-s + (2.44 + 4.23i)13-s + (−0.260 − 0.321i)15-s + 3.70·17-s − 3.66·19-s + (−0.619 + 1.61i)21-s + (3.71 + 6.42i)23-s + (2.47 − 4.28i)25-s + (4.62 − 2.36i)27-s + (−1.73 + 3.00i)29-s + (−0.358 − 0.621i)31-s + ⋯ |
L(s) = 1 | + (0.987 − 0.157i)3-s + (−0.0534 − 0.0926i)5-s + (−0.188 + 0.327i)7-s + (0.950 − 0.310i)9-s + (−0.772 + 1.33i)11-s + (0.677 + 1.17i)13-s + (−0.0673 − 0.0830i)15-s + 0.898·17-s − 0.839·19-s + (−0.135 + 0.352i)21-s + (0.773 + 1.34i)23-s + (0.494 − 0.856i)25-s + (0.890 − 0.455i)27-s + (−0.321 + 0.557i)29-s + (−0.0644 − 0.111i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.196369977\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.196369977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.71 + 0.272i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.119 + 0.207i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.56 - 4.43i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.44 - 4.23i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 + 3.66T + 19T^{2} \) |
| 23 | \( 1 + (-3.71 - 6.42i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.73 - 3.00i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.358 + 0.621i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 + (2.80 + 4.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.24 + 10.8i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.16 + 3.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.942T + 53T^{2} \) |
| 59 | \( 1 + (-3.78 - 6.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.75 - 4.77i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.330 + 0.571i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 3.66T + 73T^{2} \) |
| 79 | \( 1 + (3.11 - 5.39i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.85 + 8.40i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 7.48T + 89T^{2} \) |
| 97 | \( 1 + (-8.57 + 14.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.930199249676000893156041074650, −9.081202660317039716187030127923, −8.574453292577486638937386717073, −7.42168436179365299033962566778, −7.04014187990760149975857624584, −5.77652674658637086027353784973, −4.60779012261765606414926685487, −3.75576054487435265407405397482, −2.56424208696084448468478371086, −1.62105816068170743284440977895,
0.977744371199471365855692125168, 2.81539813650829824982620371699, 3.26166044699315201512058130211, 4.42835323597690597869210036442, 5.58615581834618453136029034650, 6.49864216357453922929306572848, 7.74235781435002959507129470103, 8.148220579321317417429000342434, 8.916652987954882668304751577397, 9.875416412019438901587905933247