| L(s) = 1 | + (−0.654 + 0.755i)2-s + (−2.71 + 1.74i)3-s + (−0.142 − 0.989i)4-s + (0.985 + 2.15i)5-s + (0.459 − 3.19i)6-s + (0.381 − 0.112i)7-s + (0.841 + 0.540i)8-s + (3.08 − 6.75i)9-s + (−2.27 − 0.668i)10-s + (2.10 + 2.42i)11-s + (2.11 + 2.44i)12-s + (−0.149 − 0.0437i)13-s + (−0.165 + 0.361i)14-s + (−6.44 − 4.14i)15-s + (−0.959 + 0.281i)16-s + (−0.467 + 3.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (−1.56 + 1.00i)3-s + (−0.0711 − 0.494i)4-s + (0.440 + 0.965i)5-s + (0.187 − 1.30i)6-s + (0.144 − 0.0423i)7-s + (0.297 + 0.191i)8-s + (1.02 − 2.25i)9-s + (−0.719 − 0.211i)10-s + (0.633 + 0.731i)11-s + (0.610 + 0.704i)12-s + (−0.0413 − 0.0121i)13-s + (−0.0441 + 0.0967i)14-s + (−1.66 − 1.06i)15-s + (−0.239 + 0.0704i)16-s + (−0.113 + 0.788i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.251304 + 0.392818i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.251304 + 0.392818i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (-1.27 + 4.62i)T \) |
| good | 3 | \( 1 + (2.71 - 1.74i)T + (1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (-0.985 - 2.15i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-0.381 + 0.112i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.10 - 2.42i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.149 + 0.0437i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.467 - 3.24i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (0.404 + 2.81i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.0538 - 0.374i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-2.31 - 1.48i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-2.66 + 5.84i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (1.66 + 3.64i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-6.25 + 4.01i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 2.97T + 47T^{2} \) |
| 53 | \( 1 + (12.5 - 3.67i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (8.29 + 2.43i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-9.37 - 6.02i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (5.50 - 6.35i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (0.233 - 0.269i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (0.802 + 5.57i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-7.23 - 2.12i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-5.56 + 12.1i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-1.81 + 1.16i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (1.09 + 2.39i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−16.31058201036686144106463126846, −15.26176219100188992713641916174, −14.46133260018117066849559915467, −12.38964002856304191860643586499, −10.99696118470398654842898526677, −10.39879290418600796774599758875, −9.280575910793844338203695457098, −6.90426959919953715696082952025, −6.03515717294458634732458133187, −4.50061879305055135564931269026,
1.26559244664599984493753511316, 5.03762002378728047362233031136, 6.35550925179304357039388360141, 7.934145780634425221753271482113, 9.531373167755924494792370376337, 11.10000032740490715108979274076, 11.84377319795394456891149450077, 12.81424752743920466867483042528, 13.70127429457734886542425611756, 16.18283636200467184563461385362
