| L(s) = 1 | + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (−0.454 − 0.231i)5-s + (−2.47 + 0.593i)7-s + (−0.453 − 0.891i)8-s + (0.157 − 0.485i)10-s + (−0.749 + 0.877i)11-s + (1.99 + 3.26i)13-s + (−0.972 − 2.34i)14-s + (0.809 − 0.587i)16-s + (−4.13 + 0.325i)17-s + (1.92 − 3.13i)19-s + (0.504 + 0.0798i)20-s + (−0.984 − 0.603i)22-s + (−6.21 − 4.51i)23-s + ⋯ |
| L(s) = 1 | + (0.110 + 0.698i)2-s + (−0.475 + 0.154i)4-s + (−0.203 − 0.103i)5-s + (−0.933 + 0.224i)7-s + (−0.160 − 0.315i)8-s + (0.0498 − 0.153i)10-s + (−0.226 + 0.264i)11-s + (0.554 + 0.905i)13-s + (−0.259 − 0.627i)14-s + (0.202 − 0.146i)16-s + (−1.00 + 0.0788i)17-s + (0.441 − 0.719i)19-s + (0.112 + 0.0178i)20-s + (−0.209 − 0.128i)22-s + (−1.29 − 0.941i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0496433 - 0.122584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0496433 - 0.122584i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.156 - 0.987i)T \) |
| 3 | \( 1 \) |
| 41 | \( 1 + (6.27 + 1.27i)T \) |
| good | 5 | \( 1 + (0.454 + 0.231i)T + (2.93 + 4.04i)T^{2} \) |
| 7 | \( 1 + (2.47 - 0.593i)T + (6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (0.749 - 0.877i)T + (-1.72 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.99 - 3.26i)T + (-5.90 + 11.5i)T^{2} \) |
| 17 | \( 1 + (4.13 - 0.325i)T + (16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (-1.92 + 3.13i)T + (-8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (6.21 + 4.51i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (9.14 + 0.719i)T + (28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (5.04 + 1.63i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.33 - 7.17i)T + (-29.9 + 21.7i)T^{2} \) |
| 43 | \( 1 + (-5.66 + 0.896i)T + (40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-1.17 + 4.91i)T + (-41.8 - 21.3i)T^{2} \) |
| 53 | \( 1 + (0.512 - 6.50i)T + (-52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (2.09 - 2.88i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.54 - 9.77i)T + (-58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (1.23 + 1.44i)T + (-10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (-8.02 - 6.85i)T + (11.1 + 70.1i)T^{2} \) |
| 73 | \( 1 + (-8.63 + 8.63i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.430 + 1.03i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 2.60iT - 83T^{2} \) |
| 89 | \( 1 + (-0.725 - 3.02i)T + (-79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (10.0 - 8.60i)T + (15.1 - 95.8i)T^{2} \) |
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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
−10.88524796810793383600053209322, −9.780438667597654367426555231804, −9.142947577385775603322524492927, −8.343539258103007168846037442594, −7.29916674581438539753184967139, −6.50317299162498247468106911372, −5.81261841340698316866517421379, −4.52453571549195646134579117564, −3.74622973451449528343320524967, −2.26254597518746666847437409117,
0.06165918314063327046986286552, 1.90251079061993747582036117784, 3.41170455360428766682426821340, 3.79916301533877631032415348396, 5.39280831716545138842964365159, 6.07509713305756707411996881068, 7.35227970619811884690663229325, 8.143250235672780622074078305420, 9.333811871252195365343712678711, 9.794369961659105099398671016306
