L(s) = 1 | + (2.45 + 0.234i)2-s + (−0.690 + 0.723i)3-s + (4.00 + 0.772i)4-s + (2.61 − 1.34i)5-s + (−1.86 + 1.61i)6-s + (−0.495 + 2.59i)7-s + (4.92 + 1.44i)8-s + (−0.0475 − 0.998i)9-s + (6.72 − 2.69i)10-s + (−0.297 − 3.12i)11-s + (−3.32 + 2.36i)12-s + (−4.04 − 0.580i)13-s + (−1.82 + 6.26i)14-s + (−0.827 + 2.81i)15-s + (4.16 + 1.66i)16-s + (−2.36 + 6.82i)17-s + ⋯ |
L(s) = 1 | + (1.73 + 0.165i)2-s + (−0.398 + 0.417i)3-s + (2.00 + 0.386i)4-s + (1.16 − 0.601i)5-s + (−0.760 + 0.659i)6-s + (−0.187 + 0.982i)7-s + (1.74 + 0.511i)8-s + (−0.0158 − 0.332i)9-s + (2.12 − 0.851i)10-s + (−0.0898 − 0.940i)11-s + (−0.959 + 0.683i)12-s + (−1.12 − 0.161i)13-s + (−0.487 + 1.67i)14-s + (−0.213 + 0.727i)15-s + (1.04 + 0.417i)16-s + (−0.572 + 1.65i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.52705 + 0.881769i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.52705 + 0.881769i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.690 - 0.723i)T \) |
| 7 | \( 1 + (0.495 - 2.59i)T \) |
| 23 | \( 1 + (-3.52 + 3.24i)T \) |
good | 2 | \( 1 + (-2.45 - 0.234i)T + (1.96 + 0.378i)T^{2} \) |
| 5 | \( 1 + (-2.61 + 1.34i)T + (2.90 - 4.07i)T^{2} \) |
| 11 | \( 1 + (0.297 + 3.12i)T + (-10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (4.04 + 0.580i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (2.36 - 6.82i)T + (-13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (2.13 + 6.17i)T + (-14.9 + 11.7i)T^{2} \) |
| 29 | \( 1 + (-5.51 - 6.36i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (0.108 - 0.0263i)T + (27.5 - 14.2i)T^{2} \) |
| 37 | \( 1 + (-2.39 + 0.114i)T + (36.8 - 3.51i)T^{2} \) |
| 41 | \( 1 + (3.67 - 5.72i)T + (-17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-0.578 - 1.97i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (7.51 + 4.33i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.83 + 4.87i)T + (-12.4 + 51.5i)T^{2} \) |
| 59 | \( 1 + (1.78 + 4.45i)T + (-42.7 + 40.7i)T^{2} \) |
| 61 | \( 1 + (-2.86 + 2.72i)T + (2.90 - 60.9i)T^{2} \) |
| 67 | \( 1 + (-4.37 - 3.11i)T + (21.9 + 63.3i)T^{2} \) |
| 71 | \( 1 + (4.18 + 9.16i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (0.476 - 2.47i)T + (-67.7 - 27.1i)T^{2} \) |
| 79 | \( 1 + (-1.71 + 2.18i)T + (-18.6 - 76.7i)T^{2} \) |
| 83 | \( 1 + (-14.7 + 9.48i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (1.77 - 7.30i)T + (-79.1 - 40.7i)T^{2} \) |
| 97 | \( 1 + (-4.47 - 2.87i)T + (40.2 + 88.2i)T^{2} \) |
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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
−11.27973585866861055704976970757, −10.46386811877355138822176254519, −9.275385394653154985270286894396, −8.461704118023174979054558683067, −6.53984035289949120662233159436, −6.22402445352245250960578165812, −5.06913360996032838659942436520, −4.86498299527906952588777746161, −3.21167814205613649807423810245, −2.18939721161427679367581665993,
1.95190828113005320438686582133, 2.85665932680255183114791023749, 4.35587748906910598304736158497, 5.11763545600270008451209429096, 6.15774203864838465509616649577, 6.90055774921241639511697271441, 7.47119377742940306301897113947, 9.683468338338922252118733236955, 10.20318494309167455230242867242, 11.18168736098979973906343830301