L(s) = 1 | + 2·2-s + (1.05 + 5.08i)3-s + 4·4-s − 6.49i·5-s + (2.10 + 10.1i)6-s − 27.4·7-s + 8·8-s + (−24.7 + 10.6i)9-s − 12.9i·10-s − 15.6·11-s + (4.20 + 20.3i)12-s − 50.8i·13-s − 54.9·14-s + (33.0 − 6.82i)15-s + 16·16-s − 87.9i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.202 + 0.979i)3-s + 0.5·4-s − 0.580i·5-s + (0.143 + 0.692i)6-s − 1.48·7-s + 0.353·8-s + (−0.918 + 0.396i)9-s − 0.410i·10-s − 0.430·11-s + (0.101 + 0.489i)12-s − 1.08i·13-s − 1.04·14-s + (0.568 − 0.117i)15-s + 0.250·16-s − 1.25i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6708763057\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6708763057\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + (-1.05 - 5.08i)T \) |
| 59 | \( 1 + (323. + 317. i)T \) |
good | 5 | \( 1 + 6.49iT - 125T^{2} \) |
| 7 | \( 1 + 27.4T + 343T^{2} \) |
| 11 | \( 1 + 15.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 50.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 87.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 54.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 138.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 24.5iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 176. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 128. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 27.5iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 121. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 353.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 249. iT - 1.48e5T^{2} \) |
| 61 | \( 1 - 441. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 148. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 259. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 229. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 4.13T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.17e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 305.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3iT - 9.12e5T^{2} \) |
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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
−10.51589212476803215903738719092, −9.977789788066581494868287994447, −9.024087172726213737206654625842, −7.986179537722385031687272211143, −6.60793726395091658136761773246, −5.56144658939871629179364195927, −4.72999006534110253475295989682, −3.49592178685444498711193614041, −2.72140706298814011971150464461, −0.16178811335001583386830835866,
1.96649922750297014069086424440, 3.03070195669944902690628854217, 4.08038768074171158724044591702, 5.99551249968589188309991357525, 6.39657278372319689408585868918, 7.25122291363943320837355062030, 8.373181377097695939059134922519, 9.567329334948022773081378935945, 10.59437879373995112128596492013, 11.57513480304506104480878794545