| L(s) = 1 | + (0.432 + 1.34i)2-s + (0.494 − 1.19i)3-s + (−1.62 + 1.16i)4-s + (−1.78 + 3.49i)5-s + (1.81 + 0.149i)6-s + (0.375 + 1.56i)7-s + (−2.27 − 1.68i)8-s + (0.942 + 0.942i)9-s + (−5.47 − 0.885i)10-s + (2.53 − 2.96i)11-s + (0.585 + 2.51i)12-s + (1.25 + 2.05i)13-s + (−1.94 + 1.18i)14-s + (3.28 + 3.84i)15-s + (1.28 − 3.78i)16-s + (−0.283 − 3.60i)17-s + ⋯ |
| L(s) = 1 | + (0.305 + 0.952i)2-s + (0.285 − 0.688i)3-s + (−0.812 + 0.582i)4-s + (−0.796 + 1.56i)5-s + (0.742 + 0.0609i)6-s + (0.142 + 0.591i)7-s + (−0.803 − 0.595i)8-s + (0.314 + 0.314i)9-s + (−1.73 − 0.280i)10-s + (0.762 − 0.893i)11-s + (0.169 + 0.725i)12-s + (0.348 + 0.568i)13-s + (−0.520 + 0.316i)14-s + (0.848 + 0.993i)15-s + (0.321 − 0.946i)16-s + (−0.0688 − 0.874i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.743628 + 0.967524i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.743628 + 0.967524i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.432 - 1.34i)T \) |
| 41 | \( 1 + (-3.29 - 5.48i)T \) |
| good | 3 | \( 1 + (-0.494 + 1.19i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (1.78 - 3.49i)T + (-2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-0.375 - 1.56i)T + (-6.23 + 3.17i)T^{2} \) |
| 11 | \( 1 + (-2.53 + 2.96i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.25 - 2.05i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (0.283 + 3.60i)T + (-16.7 + 2.65i)T^{2} \) |
| 19 | \( 1 + (-3.76 - 2.30i)T + (8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (7.12 + 5.17i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.227 + 2.88i)T + (-28.6 - 4.53i)T^{2} \) |
| 31 | \( 1 + (0.405 - 1.24i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.54 - 7.82i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.293 - 1.85i)T + (-40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.22 + 5.11i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (-6.09 - 0.479i)T + (52.3 + 8.29i)T^{2} \) |
| 59 | \( 1 + (0.993 - 1.36i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.07 + 6.77i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (2.97 - 2.53i)T + (10.4 - 66.1i)T^{2} \) |
| 71 | \( 1 + (3.90 + 3.33i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (4.47 - 4.47i)T - 73iT^{2} \) |
| 79 | \( 1 + (10.4 + 4.32i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + 5.70iT - 83T^{2} \) |
| 89 | \( 1 + (3.03 - 0.728i)T + (79.2 - 40.4i)T^{2} \) |
| 97 | \( 1 + (-3.39 + 2.90i)T + (15.1 - 95.8i)T^{2} \) |
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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
−13.60393980967277765605618702487, −12.05330583138549735138139489206, −11.53735573397305209694296444149, −10.02543622270751338383129072017, −8.542171286741094620136671945706, −7.74059183413274217946065276333, −6.83217208873664092847486197671, −6.07127347654628035734978844753, −4.16812502935407558305792726091, −2.87292736421536069994711830624,
1.24142674522898788511584887667, 3.92163525347535341448539955411, 4.16681497513008996618166865712, 5.48463857611473213978029507462, 7.63596200710747328993990867796, 8.887145655650923506572390720383, 9.502880935019189223827846433811, 10.52883094934873209316225030051, 11.78395796492352451561328632747, 12.42942308394408238775902611177
