L(s) = 1 | + (1.23 − 0.693i)2-s + (−0.793 + 0.608i)3-s + (1.03 − 1.70i)4-s + (−1.06 + 1.38i)5-s + (−0.555 + 1.30i)6-s + (−0.129 + 2.64i)7-s + (0.0962 − 2.82i)8-s + (0.258 − 0.965i)9-s + (−0.350 + 2.44i)10-s + (−0.186 + 1.41i)11-s + (0.215 + 1.98i)12-s + (−1.04 + 2.53i)13-s + (1.67 + 3.34i)14-s − 1.74i·15-s + (−1.84 − 3.55i)16-s + (5.30 + 3.06i)17-s + ⋯ |
L(s) = 1 | + (0.871 − 0.490i)2-s + (−0.458 + 0.351i)3-s + (0.519 − 0.854i)4-s + (−0.475 + 0.620i)5-s + (−0.226 + 0.530i)6-s + (−0.0490 + 0.998i)7-s + (0.0340 − 0.999i)8-s + (0.0862 − 0.321i)9-s + (−0.110 + 0.773i)10-s + (−0.0561 + 0.426i)11-s + (0.0623 + 0.573i)12-s + (−0.291 + 0.702i)13-s + (0.446 + 0.894i)14-s − 0.451i·15-s + (−0.460 − 0.887i)16-s + (1.28 + 0.742i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71839 + 0.739290i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71839 + 0.739290i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.23 + 0.693i)T \) |
| 3 | \( 1 + (0.793 - 0.608i)T \) |
| 7 | \( 1 + (0.129 - 2.64i)T \) |
good | 5 | \( 1 + (1.06 - 1.38i)T + (-1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (0.186 - 1.41i)T + (-10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (1.04 - 2.53i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (-5.30 - 3.06i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.44 + 0.848i)T + (18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (2.16 - 8.07i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (6.71 + 2.78i)T + (20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (2.87 - 4.98i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.86 + 8.94i)T + (-9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (-0.598 - 0.598i)T + 41iT^{2} \) |
| 43 | \( 1 + (1.65 - 0.687i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-1.37 + 0.793i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.46 + 11.1i)T + (-51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (-2.07 - 0.273i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.482 - 3.66i)T + (-58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (-2.74 + 2.10i)T + (17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (5.98 - 5.98i)T - 71iT^{2} \) |
| 73 | \( 1 + (-12.4 + 3.34i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-14.3 + 8.25i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.28 - 3.11i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (13.6 + 3.65i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + 4.06T + 97T^{2} \) |
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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
−10.96965465068767311664944117466, −9.694913345427498446465794841700, −9.483437794226354619815259033712, −7.71004092405295443249501693427, −6.96000106397242342901566614181, −5.65368430444196609704531049032, −5.36862090279802662830886063916, −3.91341765435281195154261292576, −3.20593334961140009526386911502, −1.77938697762762546698195418984,
0.838402755987290374808327291371, 2.93680458766906451794319897451, 4.01975942738207522869444567913, 4.99871935085276299349699720499, 5.73853070965126040031982131099, 6.84077341857807079593049344280, 7.77015262829375006375945993804, 8.049265500396400304747040191655, 9.582349371034297135955877488525, 10.63357174867581641255996872363