L(s) = 1 | + (−1.68 + 0.396i)3-s + 5-s + (2 − 1.73i)7-s + (2.68 − 1.33i)9-s + 0.792i·11-s + 5.84i·13-s + (−1.68 + 0.396i)15-s − 1.37·17-s + 3.46i·19-s + (−2.68 + 3.71i)21-s + 1.87i·23-s + 25-s + (−4 + 3.31i)27-s − 4.25i·29-s − 3.46i·31-s + ⋯ |
L(s) = 1 | + (−0.973 + 0.228i)3-s + 0.447·5-s + (0.755 − 0.654i)7-s + (0.895 − 0.445i)9-s + 0.238i·11-s + 1.61i·13-s + (−0.435 + 0.102i)15-s − 0.332·17-s + 0.794i·19-s + (−0.586 + 0.810i)21-s + 0.391i·23-s + 0.200·25-s + (−0.769 + 0.638i)27-s − 0.790i·29-s − 0.622i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.378963161\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.378963161\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.68 - 0.396i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 11 | \( 1 - 0.792iT - 11T^{2} \) |
| 13 | \( 1 - 5.84iT - 13T^{2} \) |
| 17 | \( 1 + 1.37T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 1.87iT - 23T^{2} \) |
| 29 | \( 1 + 4.25iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6.74T + 43T^{2} \) |
| 47 | \( 1 - 7.37T + 47T^{2} \) |
| 53 | \( 1 - 8.51iT - 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 6.74T + 67T^{2} \) |
| 71 | \( 1 - 13.5iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 3.37T + 79T^{2} \) |
| 83 | \( 1 + 5.48T + 83T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 97 | \( 1 + 1.08iT - 97T^{2} \) |
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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
−9.643538803368506647547323381244, −8.838689203430133646601579291428, −7.67930955369150265876515477058, −7.00819996519521335585037228765, −6.19580106160943488672759150192, −5.43541688624429952774164746025, −4.35352056821340304696296510715, −4.06738347388269426945894757292, −2.16337322651037326053200198931, −1.17380517002660459494909493075,
0.69300647530436120233477833381, 1.97300157660937276103089559676, 3.08276312176775623982388886195, 4.60169155985136736104765111665, 5.25909993946880658407176896038, 5.84722953507496251675542112664, 6.70384199849461612481332392893, 7.62634985092032778504849638252, 8.393520321156728289149006190428, 9.220876909162664559091098340858