L(s) = 1 | + (1.41 + 0.0447i)2-s + (−1.60 + 1.60i)3-s + (1.99 + 0.126i)4-s + (1.86 − 1.23i)5-s + (−2.34 + 2.20i)6-s + (0.275 + 0.275i)7-s + (2.81 + 0.267i)8-s − 2.17i·9-s + (2.68 − 1.66i)10-s + 0.822i·11-s + (−3.41 + 3.00i)12-s + (0.707 + 0.707i)13-s + (0.377 + 0.402i)14-s + (−1.00 + 4.98i)15-s + (3.96 + 0.504i)16-s + (−4.59 + 4.59i)17-s + ⋯ |
L(s) = 1 | + (0.999 + 0.0316i)2-s + (−0.928 + 0.928i)3-s + (0.998 + 0.0632i)4-s + (0.832 − 0.553i)5-s + (−0.957 + 0.898i)6-s + (0.104 + 0.104i)7-s + (0.995 + 0.0947i)8-s − 0.724i·9-s + (0.850 − 0.526i)10-s + 0.247i·11-s + (−0.985 + 0.867i)12-s + (0.196 + 0.196i)13-s + (0.100 + 0.107i)14-s + (−0.259 + 1.28i)15-s + (0.992 + 0.126i)16-s + (−1.11 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.81275 + 0.718655i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81275 + 0.718655i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.41 - 0.0447i)T \) |
| 5 | \( 1 + (-1.86 + 1.23i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (1.60 - 1.60i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.275 - 0.275i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.822iT - 11T^{2} \) |
| 17 | \( 1 + (4.59 - 4.59i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 + (-2.72 + 2.72i)T - 23iT^{2} \) |
| 29 | \( 1 + 7.29iT - 29T^{2} \) |
| 31 | \( 1 + 9.35iT - 31T^{2} \) |
| 37 | \( 1 + (4.29 - 4.29i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.02T + 41T^{2} \) |
| 43 | \( 1 + (4.26 - 4.26i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.16 - 3.16i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.615 - 0.615i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 1.78T + 61T^{2} \) |
| 67 | \( 1 + (7.04 + 7.04i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.71iT - 71T^{2} \) |
| 73 | \( 1 + (-0.126 - 0.126i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.97T + 79T^{2} \) |
| 83 | \( 1 + (3.04 - 3.04i)T - 83iT^{2} \) |
| 89 | \( 1 - 12.5iT - 89T^{2} \) |
| 97 | \( 1 + (3.35 - 3.35i)T - 97iT^{2} \) |
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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
−12.11190677030105156670932039751, −11.15155554400295092704602439381, −10.49349303656495631301078597023, −9.549384685739373992488336734122, −8.224661685666675219414581511724, −6.46743174501995852626223923313, −5.89332667439160838606856889336, −4.78398785924724585523879115592, −4.16740655129940546211901704464, −2.13838249307860764357920684120,
1.64481845906104782647487083813, 3.11164345451272434465483477066, 4.96607724119136644466869663877, 5.79831745100024375204688273753, 6.80876604625530860317152059492, 7.17988295831372226905668127399, 9.006645835425643225096924573074, 10.60999417568556722829779954810, 11.03013087302364409127196541943, 12.03019819657889687102441303972