L(s) = 1 | + (−0.107 + 0.148i)2-s + (−1.39 + 0.454i)3-s + (0.607 + 1.87i)4-s + (0.0833 − 0.256i)6-s − 3.26i·7-s + (−0.690 − 0.224i)8-s + (−0.674 + 0.489i)9-s + (−1.61 − 1.17i)11-s + (−1.70 − 2.34i)12-s + (−0.174 − 0.239i)13-s + (0.483 + 0.351i)14-s + (−3.07 + 2.23i)16-s + (−4.91 − 1.59i)17-s − 0.152i·18-s + (−0.534 + 1.64i)19-s + ⋯ |
L(s) = 1 | + (−0.0761 + 0.104i)2-s + (−0.808 + 0.262i)3-s + (0.303 + 0.935i)4-s + (0.0340 − 0.104i)6-s − 1.23i·7-s + (−0.244 − 0.0793i)8-s + (−0.224 + 0.163i)9-s + (−0.487 − 0.354i)11-s + (−0.491 − 0.675i)12-s + (−0.0483 − 0.0665i)13-s + (0.129 + 0.0938i)14-s + (−0.768 + 0.558i)16-s + (−1.19 − 0.386i)17-s − 0.0359i·18-s + (−0.122 + 0.377i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.143336 - 0.225861i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.143336 - 0.225861i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (0.107 - 0.148i)T + (-0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (1.39 - 0.454i)T + (2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + 3.26iT - 7T^{2} \) |
| 11 | \( 1 + (1.61 + 1.17i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.174 + 0.239i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (4.91 + 1.59i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.534 - 1.64i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (0.516 - 0.711i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.82 + 5.62i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.88 + 5.80i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (4.75 + 6.54i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.821 + 0.596i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 3.24iT - 43T^{2} \) |
| 47 | \( 1 + (4.01 - 1.30i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (7.70 - 2.50i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.80 + 3.48i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.740 + 0.538i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-6.55 - 2.12i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (1.84 + 5.67i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (5.19 - 7.14i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.39 - 7.38i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (13.8 + 4.48i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.08 - 4.42i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.39 + 2.07i)T + (78.4 - 57.0i)T^{2} \) |
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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
−10.61459597030078652136298809301, −9.567220334301982380755104538064, −8.352952590623481374160610766693, −7.67482149901314812835848886855, −6.78715235677889479473332422829, −5.88082048081547192359891099093, −4.62465838329321984475709419802, −3.81590858635048340934948757747, −2.45577024393013796194616213774, −0.15459992459829442774747433099,
1.71873469978833849989652064776, 2.86196695945995320566698408567, 4.85595445568574415413349434392, 5.45090542808218883917386457405, 6.36665756855456053406474977365, 6.92295944783999293192582054429, 8.544296667620770209078006714452, 9.108861288039614094039913382108, 10.21627174010747110968262087475, 10.95435362262085235434396006899