L(s) = 1 | + (0.891 − 0.453i)5-s + (0.142 − 0.896i)13-s + (−0.734 − 1.44i)17-s + (0.587 − 0.809i)25-s + (−0.0966 − 0.297i)29-s + (−0.309 − 0.0489i)37-s + (−0.533 − 0.734i)41-s + i·49-s + (0.280 − 0.550i)53-s + (1.53 + 1.11i)61-s + (−0.280 − 0.863i)65-s + (1.76 − 0.278i)73-s + (−1.30 − 0.951i)85-s + (−1.59 − 1.16i)89-s + (−0.896 + 1.76i)97-s + ⋯ |
L(s) = 1 | + (0.891 − 0.453i)5-s + (0.142 − 0.896i)13-s + (−0.734 − 1.44i)17-s + (0.587 − 0.809i)25-s + (−0.0966 − 0.297i)29-s + (−0.309 − 0.0489i)37-s + (−0.533 − 0.734i)41-s + i·49-s + (0.280 − 0.550i)53-s + (1.53 + 1.11i)61-s + (−0.280 − 0.863i)65-s + (1.76 − 0.278i)73-s + (−1.30 − 0.951i)85-s + (−1.59 − 1.16i)89-s + (−0.896 + 1.76i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.426323521\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.426323521\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.891 + 0.453i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
| 17 | \( 1 + (0.734 + 1.44i)T + (-0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 29 | \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.0489i)T + (0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (0.533 + 0.734i)T + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (-0.280 + 0.550i)T + (-0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (1.59 + 1.16i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
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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
−8.725203491430898459150574364922, −7.927669684265896303840742269391, −7.05620237490325613052502035677, −6.36893170397519025973321325506, −5.43788009067373811705047696565, −5.04587676157798213081363027539, −4.02258830871040513746725860113, −2.88514149122414953238037386621, −2.13895075802270648919704488491, −0.840549776385775673521892692054,
1.58374008495254763340833843470, 2.24611879573413677463822039234, 3.38425102285641177910995326550, 4.21579471916684780342161448997, 5.16596030562869434501174238335, 5.99421700569432475280145222617, 6.61997355140169125346538855463, 7.14226005033902694244753120153, 8.352179755662939315155535000338, 8.748371980343627743698697385259