L(s) = 1 | + (−1.41 + 0.0788i)2-s + (−1.63 + 1.63i)3-s + (1.98 − 0.222i)4-s + (−0.963 − 2.01i)5-s + (2.17 − 2.43i)6-s + (3.04 + 3.04i)7-s + (−2.78 + 0.470i)8-s − 2.33i·9-s + (1.51 + 2.77i)10-s + 0.497i·11-s + (−2.88 + 3.60i)12-s + (0.707 + 0.707i)13-s + (−4.54 − 4.06i)14-s + (4.86 + 1.72i)15-s + (3.90 − 0.884i)16-s + (−2.41 + 2.41i)17-s + ⋯ |
L(s) = 1 | + (−0.998 + 0.0557i)2-s + (−0.942 + 0.942i)3-s + (0.993 − 0.111i)4-s + (−0.430 − 0.902i)5-s + (0.888 − 0.993i)6-s + (1.15 + 1.15i)7-s + (−0.986 + 0.166i)8-s − 0.777i·9-s + (0.480 + 0.876i)10-s + 0.149i·11-s + (−0.832 + 1.04i)12-s + (0.196 + 0.196i)13-s + (−1.21 − 1.08i)14-s + (1.25 + 0.444i)15-s + (0.975 − 0.221i)16-s + (−0.585 + 0.585i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.168116 + 0.407885i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.168116 + 0.407885i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 - 0.0788i)T \) |
| 5 | \( 1 + (0.963 + 2.01i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (1.63 - 1.63i)T - 3iT^{2} \) |
| 7 | \( 1 + (-3.04 - 3.04i)T + 7iT^{2} \) |
| 11 | \( 1 - 0.497iT - 11T^{2} \) |
| 17 | \( 1 + (2.41 - 2.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.62T + 19T^{2} \) |
| 23 | \( 1 + (4.41 - 4.41i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.13iT - 29T^{2} \) |
| 31 | \( 1 - 3.14iT - 31T^{2} \) |
| 37 | \( 1 + (-4.40 + 4.40i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.49T + 41T^{2} \) |
| 43 | \( 1 + (5.16 - 5.16i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.110 - 0.110i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.203 - 0.203i)T + 53iT^{2} \) |
| 59 | \( 1 - 7.36T + 59T^{2} \) |
| 61 | \( 1 + 6.43T + 61T^{2} \) |
| 67 | \( 1 + (-7.40 - 7.40i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.618iT - 71T^{2} \) |
| 73 | \( 1 + (11.5 + 11.5i)T + 73iT^{2} \) |
| 79 | \( 1 - 2.69T + 79T^{2} \) |
| 83 | \( 1 + (-5.93 + 5.93i)T - 83iT^{2} \) |
| 89 | \( 1 - 12.7iT - 89T^{2} \) |
| 97 | \( 1 + (-8.15 + 8.15i)T - 97iT^{2} \) |
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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
−11.87297954224133819244275529652, −11.28819522893380197805050459455, −10.57247858345058964601649685859, −9.344938692008502493000468633905, −8.647105580782071901579463709805, −7.84438023544341247888881133007, −6.13346073877390701601760546948, −5.28975693638546555931890132761, −4.24260997598854481303996848466, −1.85834868544381851834580521351,
0.52241506492675401546837511837, 2.17595870875327261197273031389, 4.20061917964375469934322567879, 6.10025572636240126071667894260, 6.83003069224029905580534261131, 7.65061081206718815427887382494, 8.331230540886836037867842591870, 10.07592124825778736723925279156, 10.94795586807295589276349035011, 11.30143235941395072073512923025