L(s) = 1 | + (0.222 − 0.974i)2-s + (1.37 − 1.72i)3-s + (−0.900 − 0.433i)4-s + (−0.294 + 0.369i)5-s + (−1.37 − 1.72i)6-s + (−0.972 + 2.46i)7-s + (−0.623 + 0.781i)8-s + (−0.412 − 1.80i)9-s + (0.294 + 0.369i)10-s + (−0.105 + 0.461i)11-s + (−1.98 + 0.955i)12-s + (0.0313 − 0.137i)13-s + (2.18 + 1.49i)14-s + (0.231 + 1.01i)15-s + (0.623 + 0.781i)16-s + (6.09 − 2.93i)17-s + ⋯ |
L(s) = 1 | + (0.157 − 0.689i)2-s + (0.792 − 0.994i)3-s + (−0.450 − 0.216i)4-s + (−0.131 + 0.165i)5-s + (−0.560 − 0.703i)6-s + (−0.367 + 0.929i)7-s + (−0.220 + 0.276i)8-s + (−0.137 − 0.602i)9-s + (0.0932 + 0.116i)10-s + (−0.0317 + 0.139i)11-s + (−0.572 + 0.275i)12-s + (0.00869 − 0.0380i)13-s + (0.583 + 0.399i)14-s + (0.0598 + 0.262i)15-s + (0.155 + 0.195i)16-s + (1.47 − 0.711i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.934359 - 0.782153i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.934359 - 0.782153i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.222 + 0.974i)T \) |
| 7 | \( 1 + (0.972 - 2.46i)T \) |
good | 3 | \( 1 + (-1.37 + 1.72i)T + (-0.667 - 2.92i)T^{2} \) |
| 5 | \( 1 + (0.294 - 0.369i)T + (-1.11 - 4.87i)T^{2} \) |
| 11 | \( 1 + (0.105 - 0.461i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.0313 + 0.137i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-6.09 + 2.93i)T + (10.5 - 13.2i)T^{2} \) |
| 19 | \( 1 + 4.64T + 19T^{2} \) |
| 23 | \( 1 + (3.73 + 1.79i)T + (14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (4.42 - 2.12i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 - 0.381T + 31T^{2} \) |
| 37 | \( 1 + (9.73 - 4.68i)T + (23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + (-6.40 + 8.03i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (4.50 + 5.64i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-0.623 + 2.73i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-11.3 - 5.44i)T + (33.0 + 41.4i)T^{2} \) |
| 59 | \( 1 + (7.83 + 9.83i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (-3.49 + 1.68i)T + (38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + 8.63T + 67T^{2} \) |
| 71 | \( 1 + (2.99 + 1.44i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-1.41 - 6.19i)T + (-65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 - 9.98T + 79T^{2} \) |
| 83 | \( 1 + (-2.43 - 10.6i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-0.676 - 2.96i)T + (-80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 + 3.36T + 97T^{2} \) |
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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
−13.56464231774992285738322600824, −12.52508761579754992646672198514, −12.02740911273869323864375590991, −10.51736761264115408564108004986, −9.234352785605502717926741251106, −8.267819513783762700965256303906, −7.03761841582629381276304504509, −5.46920846853225363353823427997, −3.35017140725680172569522535881, −2.06781795112045790270722148753,
3.50845291057178766919656317037, 4.39380538502797712636194103151, 6.09421370422275820207113235167, 7.63193887832852064229082620973, 8.601531882493345924051601939234, 9.793674563937412800596168130954, 10.52097124740080675066425507504, 12.30489794140234708173830080281, 13.45872898666953022906301023486, 14.43260432640949790549049424676