| L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.913 − 0.406i)3-s + (0.309 − 0.951i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + (0.955 + 1.06i)7-s + (0.309 + 0.951i)8-s + (0.669 − 0.743i)9-s + (0.913 + 0.406i)10-s + (−4.33 + 0.920i)11-s + (−0.104 − 0.994i)12-s + (0.559 − 5.32i)13-s + (−1.39 − 0.297i)14-s + (−0.809 − 0.587i)15-s + (−0.809 − 0.587i)16-s + (5.84 + 1.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.527 − 0.234i)3-s + (0.154 − 0.475i)4-s + (−0.223 − 0.387i)5-s + (−0.204 + 0.353i)6-s + (0.361 + 0.401i)7-s + (0.109 + 0.336i)8-s + (0.223 − 0.247i)9-s + (0.288 + 0.128i)10-s + (−1.30 + 0.277i)11-s + (−0.0301 − 0.287i)12-s + (0.155 − 1.47i)13-s + (−0.373 − 0.0793i)14-s + (−0.208 − 0.151i)15-s + (−0.202 − 0.146i)16-s + (1.41 + 0.301i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06997 - 0.597973i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06997 - 0.597973i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (-5.13 + 2.14i)T \) |
| good | 7 | \( 1 + (-0.955 - 1.06i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (4.33 - 0.920i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.559 + 5.32i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-5.84 - 1.24i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.590 + 5.61i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.413 + 1.27i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.774 + 0.562i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (3.16 - 5.48i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.41 - 1.07i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (0.908 + 8.64i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (9.73 + 7.07i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.81 + 2.01i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-2.83 + 1.26i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 + (6.36 + 11.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.74 - 3.05i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (-3.77 + 0.802i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-12.9 - 2.76i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (4.81 + 2.14i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (2.77 - 8.53i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (2.63 - 8.12i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.01418362658305264381219584944, −8.817429618582982002386352390795, −8.081860031209679123545513760555, −7.84047849398792119058542050674, −6.71894215144659544422351625764, −5.46548759255565622749004409234, −4.97705569115661804201861366421, −3.31492013951612604630174901559, −2.28276566316649468050622378664, −0.68727882237931437062567542501,
1.48442150696972292746450852978, 2.75161887219035385344592072352, 3.67035794898093721820774967950, 4.67223919531086976167656924807, 5.94405460122060412828637233255, 7.19379698616192979643914450121, 7.86221399953206646650638739209, 8.431409700543896744547779301033, 9.553664262712696104892654799125, 10.14030434014492392434293189566
