L(s) = 1 | + (1.15 − 1.99i)5-s + (−0.271 + 2.63i)7-s + (−2.23 − 3.86i)11-s − 2.84·13-s + (−0.115 − 0.199i)17-s + (−1.49 + 2.58i)19-s + (0.400 − 0.693i)23-s + (−0.149 − 0.259i)25-s − 7.65·29-s + (−2.64 − 4.57i)31-s + (4.93 + 3.57i)35-s + (−1.69 + 2.93i)37-s + 1.79·41-s − 9.71·43-s + (−2.88 + 5.00i)47-s + ⋯ |
L(s) = 1 | + (0.514 − 0.891i)5-s + (−0.102 + 0.994i)7-s + (−0.672 − 1.16i)11-s − 0.788·13-s + (−0.0279 − 0.0484i)17-s + (−0.342 + 0.593i)19-s + (0.0834 − 0.144i)23-s + (−0.0299 − 0.0518i)25-s − 1.42·29-s + (−0.474 − 0.822i)31-s + (0.833 + 0.603i)35-s + (−0.278 + 0.482i)37-s + 0.281·41-s − 1.48·43-s + (−0.421 + 0.729i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2290542958\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2290542958\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.271 - 2.63i)T \) |
good | 5 | \( 1 + (-1.15 + 1.99i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.23 + 3.86i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.84T + 13T^{2} \) |
| 17 | \( 1 + (0.115 + 0.199i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.49 - 2.58i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.400 + 0.693i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.65T + 29T^{2} \) |
| 31 | \( 1 + (2.64 + 4.57i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.69 - 2.93i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.79T + 41T^{2} \) |
| 43 | \( 1 + 9.71T + 43T^{2} \) |
| 47 | \( 1 + (2.88 - 5.00i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.31 + 7.48i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.17 - 7.23i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.58 + 11.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.76 - 6.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.59T + 71T^{2} \) |
| 73 | \( 1 + (2.29 + 3.97i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.83 - 8.37i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + (-0.944 + 1.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.423088466605878623364928638770, −8.284227026473832809445386756865, −7.07662663692545463291911308549, −5.99522029124212094992835933658, −5.48391121837262293162093863415, −4.90300385774432608055954725429, −3.62473643282423574710410953420, −2.60762765337538054376137315698, −1.66138324372166694391298773249, −0.07095593170571036653196875849,
1.78940804018938606819348666607, 2.64334205560068950037603983677, 3.66591363258143328174741874768, 4.66423632752403765121193968890, 5.39257997061032931824668749846, 6.56044863184091969536783351590, 7.14894908167432278266746624962, 7.53114396372843828412883282803, 8.657007217543784247272891671808, 9.818345672549777101388384635054