| 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
−12.45463657754160389365834061808, −12.01515006787481196270040910433, −10.47235610955931521843396839945, −10.28965453324529096907855010665, −8.374323623839307995155207830043, −7.68338130898952311562059691451, −6.56258891045187389422702591790, −5.72631360520467479146958922593, −4.57424221075123793260896264970, −2.84583974136973075078289341902,
0.30139880360704918007268663098, 2.52018323025141028767296120778, 4.55644480568727246337787006895, 4.91205141088746895022971946665, 6.38271662682031587406209399790, 7.940389009398849476817471122386, 8.898802924542511967959889135484, 9.850879898336991953047994031988, 11.17886622050268665888398003311, 11.52492186736759723993176166800
