| L(s) = 1 | + (−0.654 − 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (−1.70 + 3.73i)5-s + (−0.142 − 0.989i)6-s + (−0.959 − 0.281i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (3.94 − 1.15i)10-s + (−0.271 + 0.313i)11-s + (−0.654 + 0.755i)12-s + (−5.88 + 1.72i)13-s + (0.415 + 0.909i)14-s + (−3.45 + 2.22i)15-s + (−0.959 − 0.281i)16-s + (−0.143 − 0.996i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (0.485 + 0.312i)3-s + (−0.0711 + 0.494i)4-s + (−0.763 + 1.67i)5-s + (−0.0580 − 0.404i)6-s + (−0.362 − 0.106i)7-s + (0.297 − 0.191i)8-s + (0.138 + 0.303i)9-s + (1.24 − 0.366i)10-s + (−0.0817 + 0.0943i)11-s + (−0.189 + 0.218i)12-s + (−1.63 + 0.479i)13-s + (0.111 + 0.243i)14-s + (−0.892 + 0.573i)15-s + (−0.239 − 0.0704i)16-s + (−0.0347 − 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0156842 - 0.282852i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0156842 - 0.282852i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 3 | \( 1 + (-0.841 - 0.540i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (-4.79 - 0.209i)T \) |
| good | 5 | \( 1 + (1.70 - 3.73i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (0.271 - 0.313i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (5.88 - 1.72i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.143 + 0.996i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.459 + 3.19i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.184 - 1.28i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (1.55 - 1.00i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (1.80 + 3.94i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-3.70 + 8.10i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.65 + 1.06i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 0.672T + 47T^{2} \) |
| 53 | \( 1 + (9.52 + 2.79i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (8.82 - 2.59i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (9.40 - 6.04i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (2.15 + 2.48i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-6.42 - 7.41i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.03 + 7.21i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (6.90 - 2.02i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-4.03 - 8.84i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-12.0 - 7.71i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (7.63 - 16.7i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.59439037492467848075802665348, −9.626035324752065660144630194423, −9.083114674653984717047728194286, −7.76113039333724548269517649345, −7.25845354109270181408637404684, −6.66181175391681015899323181842, −4.94685640061127177344391057575, −3.87002728692492151871946826135, −2.97258329435949819172029345929, −2.36796115293318488607509546341,
0.14237966910356135437529527882, 1.49645816431202489931992072966, 3.11449647436348850653162155044, 4.50611867361494727123971538241, 5.11407555295373874600617736741, 6.22261499502087589960145641094, 7.51143755991446290701625906293, 7.85228754207889868001406432424, 8.676086766966767654182112129616, 9.384770127385008413841855672272
