| L(s) = 1 | + (0.0713 − 0.997i)2-s + (−1.35 + 0.293i)3-s + (−0.989 − 0.142i)4-s + (−0.868 − 2.06i)5-s + (0.196 + 1.36i)6-s + (0.593 − 0.324i)7-s + (−0.212 + 0.977i)8-s + (−0.992 + 0.453i)9-s + (−2.11 + 0.718i)10-s + (−2.95 − 2.55i)11-s + (1.37 − 0.0985i)12-s + (−2.97 + 5.45i)13-s + (−0.280 − 0.615i)14-s + (1.77 + 2.52i)15-s + (0.959 + 0.281i)16-s + (−4.63 − 3.47i)17-s + ⋯ |
| L(s) = 1 | + (0.0504 − 0.705i)2-s + (−0.779 + 0.169i)3-s + (−0.494 − 0.0711i)4-s + (−0.388 − 0.921i)5-s + (0.0802 + 0.558i)6-s + (0.224 − 0.122i)7-s + (−0.0751 + 0.345i)8-s + (−0.330 + 0.151i)9-s + (−0.669 + 0.227i)10-s + (−0.890 − 0.771i)11-s + (0.397 − 0.0284i)12-s + (−0.826 + 1.51i)13-s + (−0.0750 − 0.164i)14-s + (0.458 + 0.652i)15-s + (0.239 + 0.0704i)16-s + (−1.12 − 0.841i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0344288 + 0.355926i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0344288 + 0.355926i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0713 + 0.997i)T \) |
| 5 | \( 1 + (0.868 + 2.06i)T \) |
| 23 | \( 1 + (4.71 - 0.889i)T \) |
| good | 3 | \( 1 + (1.35 - 0.293i)T + (2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-0.593 + 0.324i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (2.95 + 2.55i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.97 - 5.45i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (4.63 + 3.47i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-0.740 + 5.15i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-7.94 + 1.14i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-7.58 + 4.87i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.301i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (0.603 - 1.32i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.776 + 3.56i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-4.53 - 4.53i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.83 + 8.86i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (2.82 + 9.60i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (4.29 + 6.67i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (5.82 + 0.416i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (4.01 + 4.63i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-7.42 - 9.91i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (5.43 - 1.59i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-2.45 + 6.58i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-6.82 - 4.38i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (1.15 + 3.10i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−11.52492186736759723993176166800, −11.17886622050268665888398003311, −9.850879898336991953047994031988, −8.898802924542511967959889135484, −7.940389009398849476817471122386, −6.38271662682031587406209399790, −4.91205141088746895022971946665, −4.55644480568727246337787006895, −2.52018323025141028767296120778, −0.30139880360704918007268663098,
2.84583974136973075078289341902, 4.57424221075123793260896264970, 5.72631360520467479146958922593, 6.56258891045187389422702591790, 7.68338130898952311562059691451, 8.374323623839307995155207830043, 10.28965453324529096907855010665, 10.47235610955931521843396839945, 12.01515006787481196270040910433, 12.45463657754160389365834061808
