| L(s) = 1 | + (−0.995 − 0.0950i)2-s + (−0.690 + 0.723i)3-s + (0.981 + 0.189i)4-s + (−2.11 + 1.08i)5-s + (0.755 − 0.654i)6-s + (0.852 − 2.50i)7-s + (−0.959 − 0.281i)8-s + (−0.0475 − 0.998i)9-s + (2.20 − 0.883i)10-s + (0.0226 + 0.237i)11-s + (−0.814 + 0.580i)12-s + (2.58 + 0.371i)13-s + (−1.08 + 2.41i)14-s + (0.669 − 2.28i)15-s + (0.928 + 0.371i)16-s + (−2.25 + 6.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.703 − 0.0672i)2-s + (−0.398 + 0.417i)3-s + (0.490 + 0.0946i)4-s + (−0.945 + 0.487i)5-s + (0.308 − 0.267i)6-s + (0.322 − 0.946i)7-s + (−0.339 − 0.0996i)8-s + (−0.0158 − 0.332i)9-s + (0.697 − 0.279i)10-s + (0.00683 + 0.0715i)11-s + (−0.235 + 0.167i)12-s + (0.717 + 0.103i)13-s + (−0.290 + 0.644i)14-s + (0.172 − 0.588i)15-s + (0.232 + 0.0929i)16-s + (−0.546 + 1.57i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0573602 + 0.281283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0573602 + 0.281283i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.995 + 0.0950i)T \) |
| 3 | \( 1 + (0.690 - 0.723i)T \) |
| 7 | \( 1 + (-0.852 + 2.50i)T \) |
| 23 | \( 1 + (-4.77 - 0.495i)T \) |
| good | 5 | \( 1 + (2.11 - 1.08i)T + (2.90 - 4.07i)T^{2} \) |
| 11 | \( 1 + (-0.0226 - 0.237i)T + (-10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (-2.58 - 0.371i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (2.25 - 6.51i)T + (-13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (1.71 + 4.94i)T + (-14.9 + 11.7i)T^{2} \) |
| 29 | \( 1 + (-3.12 - 3.60i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (7.33 - 1.77i)T + (27.5 - 14.2i)T^{2} \) |
| 37 | \( 1 + (2.38 - 0.113i)T + (36.8 - 3.51i)T^{2} \) |
| 41 | \( 1 + (2.56 - 3.98i)T + (-17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-2.13 - 7.26i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (9.49 + 5.47i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.19 + 7.88i)T + (-12.4 + 51.5i)T^{2} \) |
| 59 | \( 1 + (5.15 + 12.8i)T + (-42.7 + 40.7i)T^{2} \) |
| 61 | \( 1 + (6.34 - 6.05i)T + (2.90 - 60.9i)T^{2} \) |
| 67 | \( 1 + (0.169 + 0.120i)T + (21.9 + 63.3i)T^{2} \) |
| 71 | \( 1 + (-6.37 - 13.9i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.912 + 4.73i)T + (-67.7 - 27.1i)T^{2} \) |
| 79 | \( 1 + (10.1 - 12.8i)T + (-18.6 - 76.7i)T^{2} \) |
| 83 | \( 1 + (0.362 - 0.233i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (1.22 - 5.04i)T + (-79.1 - 40.7i)T^{2} \) |
| 97 | \( 1 + (2.39 + 1.54i)T + (40.2 + 88.2i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.66967978505944260842311222395, −9.627511745822932300630523097095, −8.617882363085194551957151058557, −8.015138337886390089796081778594, −6.92841008065685662349536213774, −6.54893647866696139540155997551, −5.03126495942930552552674831672, −4.01341538671881087795668416757, −3.27491034345044448953437589339, −1.44846942108391499865197049462,
0.18639182611902857045746236527, 1.66658792881896480980391598486, 3.01283993838136534172135812593, 4.43289282140639136753075056111, 5.43294577611594198070651272900, 6.30573964309002634969613206088, 7.34621451839144431593800016955, 8.027198175875647476599766257377, 8.772363714627480256632716440620, 9.366695903814888574307677953733
