L(s) = 1 | + (1 − 1.73i)5-s + (1.5 − 2.59i)9-s + (1 + 1.73i)11-s + 13-s + (3 + 5.19i)17-s + (−3 + 5.19i)19-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s + 2·29-s + (5 + 8.66i)31-s + (3 − 5.19i)37-s + 6·41-s + 4·43-s + (−3 − 5.19i)45-s + (−1 + 1.73i)47-s + ⋯ |
L(s) = 1 | + (0.447 − 0.774i)5-s + (0.5 − 0.866i)9-s + (0.301 + 0.522i)11-s + 0.277·13-s + (0.727 + 1.26i)17-s + (−0.688 + 1.19i)19-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s + 0.371·29-s + (0.898 + 1.55i)31-s + (0.493 − 0.854i)37-s + 0.937·41-s + 0.609·43-s + (−0.447 − 0.774i)45-s + (−0.145 + 0.252i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.118350900\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.118350900\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-5 - 8.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.010420766079092288634322711011, −8.193539058565413101320101811553, −7.52868990801572514846503236074, −6.37780283572819713656672817310, −5.98341170242348449409120938542, −5.01357180371621334163280220555, −4.03283452455296426575231494459, −3.46814096634820451459141665515, −1.81155707157280724890047392864, −1.20918338678066104249379417414,
0.806988999959568857209816037824, 2.40852956822482172293138287791, 2.77669978438021523427971363561, 4.21496948097574950025193096025, 4.80530351955309641219575176789, 5.99433791467696264493881126675, 6.46802647414106140048907437236, 7.34004442374994101776155842679, 8.041417319992677436093810672547, 8.863600638933600365605831257342