L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.14 + 1.29i)3-s − 1.00i·4-s + (0.237 − 0.237i)5-s + (0.103 + 1.72i)6-s + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (−0.359 − 2.97i)9-s − 0.335i·10-s + (2.55 + 2.55i)11-s + (1.29 + 1.14i)12-s + (3.37 + 1.25i)13-s + 1.00i·14-s + (0.0348 + 0.579i)15-s − 1.00·16-s + 5.26·17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.663 + 0.748i)3-s − 0.500i·4-s + (0.106 − 0.106i)5-s + (0.0424 + 0.705i)6-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + (−0.119 − 0.992i)9-s − 0.106i·10-s + (0.770 + 0.770i)11-s + (0.374 + 0.331i)12-s + (0.937 + 0.348i)13-s + 0.267i·14-s + (0.00900 + 0.149i)15-s − 0.250·16-s + 1.27·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.59857 + 0.156814i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59857 + 0.156814i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.14 - 1.29i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + (-3.37 - 1.25i)T \) |
good | 5 | \( 1 + (-0.237 + 0.237i)T - 5iT^{2} \) |
| 11 | \( 1 + (-2.55 - 2.55i)T + 11iT^{2} \) |
| 17 | \( 1 - 5.26T + 17T^{2} \) |
| 19 | \( 1 + (-0.906 - 0.906i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.48T + 23T^{2} \) |
| 29 | \( 1 + 7.36iT - 29T^{2} \) |
| 31 | \( 1 + (-7.20 - 7.20i)T + 31iT^{2} \) |
| 37 | \( 1 + (2.83 - 2.83i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.32 + 1.32i)T - 41iT^{2} \) |
| 43 | \( 1 - 2.75iT - 43T^{2} \) |
| 47 | \( 1 + (2.27 + 2.27i)T + 47iT^{2} \) |
| 53 | \( 1 + 8.80iT - 53T^{2} \) |
| 59 | \( 1 + (0.785 + 0.785i)T + 59iT^{2} \) |
| 61 | \( 1 - 8.74T + 61T^{2} \) |
| 67 | \( 1 + (9.68 + 9.68i)T + 67iT^{2} \) |
| 71 | \( 1 + (-4.94 + 4.94i)T - 71iT^{2} \) |
| 73 | \( 1 + (4.00 - 4.00i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.50T + 79T^{2} \) |
| 83 | \( 1 + (3.05 - 3.05i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.62 - 1.62i)T + 89iT^{2} \) |
| 97 | \( 1 + (4.86 + 4.86i)T + 97iT^{2} \) |
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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
−10.89764728762223431343138440399, −9.912716159016952821849148305933, −9.544684484645009506060827490974, −8.396621500881133484816939960042, −6.82978188385947369232974970586, −6.02129889582371286769205707141, −5.13827002456702926327949355424, −4.10787355957312896685689347527, −3.25591093242019012030873007593, −1.38271731331638374575734185294,
1.07406411149926278383208740382, 2.98376970013329652976379866795, 4.17610461015540164832671229180, 5.57608163086084874070974800894, 6.11070874338930857040683180528, 6.94694020620384051022175598828, 7.907937309830033620181316146406, 8.695910767795913786755305445172, 10.05904112292115349125150204441, 10.97424531906844250106458392828