L(s) = 1 | + (−1.41 + 0.0554i)2-s + (0.488 + 0.488i)3-s + (1.99 − 0.156i)4-s + (−0.717 − 0.663i)6-s + 4.71i·7-s + (−2.80 + 0.331i)8-s − 2.52i·9-s + (−3.91 + 3.91i)11-s + (1.05 + 0.897i)12-s + (0.0878 + 0.0878i)13-s + (−0.261 − 6.66i)14-s + (3.95 − 0.624i)16-s − 4.67·17-s + (0.139 + 3.56i)18-s + (1.81 + 1.81i)19-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.0391i)2-s + (0.282 + 0.282i)3-s + (0.996 − 0.0783i)4-s + (−0.292 − 0.270i)6-s + 1.78i·7-s + (−0.993 + 0.117i)8-s − 0.840i·9-s + (−1.17 + 1.17i)11-s + (0.303 + 0.259i)12-s + (0.0243 + 0.0243i)13-s + (−0.0698 − 1.78i)14-s + (0.987 − 0.156i)16-s − 1.13·17-s + (0.0329 + 0.840i)18-s + (0.415 + 0.415i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.342483 + 0.611351i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.342483 + 0.611351i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 - 0.0554i)T \) |
| 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
−11.58496020473706671316425684455, −10.36863504597534764541315918265, −9.625193609259052787963773091669, −8.867945025770639636617575002272, −8.242983305219029605863584589752, −7.00568519619118133135277827507, −6.04022310836646086621540276970, −4.92455247335048890182383199661, −3.00364219663611210965892883271, −2.09720915865478400325619856065,
0.57669193101252604785343834824, 2.29791616688543587884940284555, 3.63105834799326307034136316902, 5.21747116897853509816893489436, 6.66252009314043982539652557211, 7.49891523100028735595499594441, 8.062677082905319135003273473542, 9.020528458452092346273479531245, 10.34742917058952991519170345804, 10.70870381463345244793586912040