L(s) = 1 | + (0.871 − 0.871i)5-s + 7-s + (0.987 + 0.987i)11-s + (0.526 − 0.526i)13-s + 5.93i·17-s + (−2.78 − 2.78i)19-s − 8.59i·23-s + 3.48i·25-s + (5.07 + 5.07i)29-s + 7.72i·31-s + (0.871 − 0.871i)35-s + (3.36 + 3.36i)37-s + 4.55·41-s + (−1.26 + 1.26i)43-s + 5.02·47-s + ⋯ |
L(s) = 1 | + (0.389 − 0.389i)5-s + 0.377·7-s + (0.297 + 0.297i)11-s + (0.146 − 0.146i)13-s + 1.43i·17-s + (−0.638 − 0.638i)19-s − 1.79i·23-s + 0.696i·25-s + (0.942 + 0.942i)29-s + 1.38i·31-s + (0.147 − 0.147i)35-s + (0.553 + 0.553i)37-s + 0.711·41-s + (−0.193 + 0.193i)43-s + 0.733·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.184036923\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.184036923\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 5 | \( 1 + (-0.871 + 0.871i)T - 5iT^{2} \) |
| 11 | \( 1 + (-0.987 - 0.987i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.526 + 0.526i)T - 13iT^{2} \) |
| 17 | \( 1 - 5.93iT - 17T^{2} \) |
| 19 | \( 1 + (2.78 + 2.78i)T + 19iT^{2} \) |
| 23 | \( 1 + 8.59iT - 23T^{2} \) |
| 29 | \( 1 + (-5.07 - 5.07i)T + 29iT^{2} \) |
| 31 | \( 1 - 7.72iT - 31T^{2} \) |
| 37 | \( 1 + (-3.36 - 3.36i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.55T + 41T^{2} \) |
| 43 | \( 1 + (1.26 - 1.26i)T - 43iT^{2} \) |
| 47 | \( 1 - 5.02T + 47T^{2} \) |
| 53 | \( 1 + (2.07 - 2.07i)T - 53iT^{2} \) |
| 59 | \( 1 + (6.52 + 6.52i)T + 59iT^{2} \) |
| 61 | \( 1 + (-8.74 + 8.74i)T - 61iT^{2} \) |
| 67 | \( 1 + (-6.20 - 6.20i)T + 67iT^{2} \) |
| 71 | \( 1 + 1.72iT - 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 - 5.03iT - 79T^{2} \) |
| 83 | \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.97T + 89T^{2} \) |
| 97 | \( 1 + 8.71T + 97T^{2} \) |
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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
−8.532665105605052359708283948815, −7.964286043151830263968762875216, −6.75584889017869355167306965620, −6.46998847986390670528120186059, −5.42602339643805011903151123275, −4.72587397588588935770949399066, −4.04584362953504908267948932329, −2.93339675626955966958434335106, −1.93226372494471185924575314491, −1.02203397350620911878579710576,
0.74876727138663270283336031696, 2.02027537563284461840965046881, 2.77659592824631046790716210384, 3.85580203457572009349351907410, 4.55507362900217326679799405644, 5.61632193230427416115264048475, 6.07764027226863152781832496501, 6.96173350655747654080671559486, 7.69662242875976729767497038906, 8.287484233364337911196015321640