L(s) = 1 | + (0.488 + 0.488i)3-s + 4.71i·7-s − 2.52i·9-s + (3.91 − 3.91i)11-s + (−0.0878 − 0.0878i)13-s + 4.67·17-s + (−1.81 − 1.81i)19-s + (−2.30 + 2.30i)21-s − 1.63i·23-s + (2.69 − 2.69i)27-s + (3.26 + 3.26i)29-s + 2.12·31-s + 3.82·33-s + (3.97 − 3.97i)37-s − 0.0858i·39-s + ⋯ |
L(s) = 1 | + (0.282 + 0.282i)3-s + 1.78i·7-s − 0.840i·9-s + (1.17 − 1.17i)11-s + (−0.0243 − 0.0243i)13-s + 1.13·17-s + (−0.415 − 0.415i)19-s + (−0.502 + 0.502i)21-s − 0.339i·23-s + (0.519 − 0.519i)27-s + (0.606 + 0.606i)29-s + 0.382·31-s + 0.665·33-s + (0.653 − 0.653i)37-s − 0.0137i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.097589215\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.097589215\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.488 - 0.488i)T + 3iT^{2} \) |
| 7 | \( 1 - 4.71iT - 7T^{2} \) |
| 11 | \( 1 + (-3.91 + 3.91i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.0878 + 0.0878i)T + 13iT^{2} \) |
| 17 | \( 1 - 4.67T + 17T^{2} \) |
| 19 | \( 1 + (1.81 + 1.81i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.63iT - 23T^{2} \) |
| 29 | \( 1 + (-3.26 - 3.26i)T + 29iT^{2} \) |
| 31 | \( 1 - 2.12T + 31T^{2} \) |
| 37 | \( 1 + (-3.97 + 3.97i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.25iT - 41T^{2} \) |
| 43 | \( 1 + (2.27 - 2.27i)T - 43iT^{2} \) |
| 47 | \( 1 - 4.06T + 47T^{2} \) |
| 53 | \( 1 + (5.03 - 5.03i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.16 + 5.16i)T - 59iT^{2} \) |
| 61 | \( 1 + (-7.12 - 7.12i)T + 61iT^{2} \) |
| 67 | \( 1 + (-7.49 - 7.49i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.54iT - 71T^{2} \) |
| 73 | \( 1 - 8.30iT - 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 + (-1.16 - 1.16i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.24iT - 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−9.393881147531914711513507611982, −8.542287706657078764984806738035, −8.429954075830110362461188761871, −6.87028359255484887561241326218, −6.08053983889265022281233211519, −5.60252709323898754244976589188, −4.37072971429195395125429295413, −3.30815303181255664091178636470, −2.66783689209466070300578389930, −1.10943217093028891605559652026,
1.05282916458530168937109462239, 2.02390282507104762117602913218, 3.52127244197279911933507752531, 4.23164967643466807769989687449, 5.02081237849516183469713486031, 6.37326281869284465356307631283, 7.10805210009860595935619074056, 7.64838581338570526546060797810, 8.330373338603847385460989195601, 9.560754501832947398479409934208