L(s) = 1 | + (0.707 + 0.707i)2-s + 1.00i·4-s + (−1.25 + 1.84i)5-s + (−0.707 + 0.707i)8-s + (−2.19 + 0.417i)10-s + 6.26i·11-s + (−4.39 − 4.39i)13-s − 1.00·16-s + (−2.28 − 2.28i)17-s + 0.722i·19-s + (−1.84 − 1.25i)20-s + (−4.42 + 4.42i)22-s + (−0.707 + 0.707i)23-s + (−1.83 − 4.65i)25-s − 6.21i·26-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.562 + 0.826i)5-s + (−0.250 + 0.250i)8-s + (−0.694 + 0.132i)10-s + 1.88i·11-s + (−1.21 − 1.21i)13-s − 0.250·16-s + (−0.553 − 0.553i)17-s + 0.165i·19-s + (−0.413 − 0.281i)20-s + (−0.944 + 0.944i)22-s + (−0.147 + 0.147i)23-s + (−0.367 − 0.930i)25-s − 1.21i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3701817742\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3701817742\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.25 - 1.84i)T \) |
| 23 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 6.26iT - 11T^{2} \) |
| 13 | \( 1 + (4.39 + 4.39i)T + 13iT^{2} \) |
| 17 | \( 1 + (2.28 + 2.28i)T + 17iT^{2} \) |
| 19 | \( 1 - 0.722iT - 19T^{2} \) |
| 29 | \( 1 - 5.95T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + (-1.05 + 1.05i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.80iT - 41T^{2} \) |
| 43 | \( 1 + (-1.20 - 1.20i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.78 + 8.78i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.23 - 2.23i)T - 53iT^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 4.90T + 61T^{2} \) |
| 67 | \( 1 + (7.15 - 7.15i)T - 67iT^{2} \) |
| 71 | \( 1 - 3.68iT - 71T^{2} \) |
| 73 | \( 1 + (11.6 + 11.6i)T + 73iT^{2} \) |
| 79 | \( 1 - 0.442iT - 79T^{2} \) |
| 83 | \( 1 + (11.9 - 11.9i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.95T + 89T^{2} \) |
| 97 | \( 1 + (-6.97 + 6.97i)T - 97iT^{2} \) |
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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
−9.884096438183772477962017292711, −8.677920192107306677621812415207, −7.69461702780691572321689022649, −7.25097019501879534336291433465, −6.78138794560370870402913515593, −5.58578506291444627166381720482, −4.79271169946250851928965454556, −4.08979637935471070499078977535, −2.95659758246171932118936944117, −2.20902422635025912448030910039,
0.10660761960426614525355710119, 1.43089226952879790182066216369, 2.69155319331269731288877005045, 3.69403845814619855395787004696, 4.46971159354111376194940720493, 5.17771001333524734834877253909, 6.11487224773346195586244485799, 6.90534612203940278607801227996, 8.028001964578716124049762172556, 8.673985212935900510601749558426