L(s) = 1 | − 8.75i·2-s + (10.6 + 11.3i)3-s − 44.7·4-s − 43.1·5-s + (99.7 − 93.2i)6-s − 151. i·7-s + 111. i·8-s + (−16.4 + 242. i)9-s + 378. i·10-s − 610.·11-s + (−476. − 509. i)12-s + 132.·13-s − 1.32e3·14-s + (−459. − 491. i)15-s − 454.·16-s − 1.72e3·17-s + ⋯ |
L(s) = 1 | − 1.54i·2-s + (0.682 + 0.730i)3-s − 1.39·4-s − 0.772·5-s + (1.13 − 1.05i)6-s − 1.16i·7-s + 0.615i·8-s + (−0.0676 + 0.997i)9-s + 1.19i·10-s − 1.52·11-s + (−0.954 − 1.02i)12-s + 0.217·13-s − 1.80·14-s + (−0.527 − 0.564i)15-s − 0.444·16-s − 1.44·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.246682 + 0.587979i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.246682 + 0.587979i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-10.6 - 11.3i)T \) |
| 23 | \( 1 + (-2.53e3 + 62.8i)T \) |
good | 2 | \( 1 + 8.75iT - 32T^{2} \) |
| 5 | \( 1 + 43.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 151. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 610.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 132.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.72e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 554. iT - 2.47e6T^{2} \) |
| 29 | \( 1 + 7.00e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.32e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 231. iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.84e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.17e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.89e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.71e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.20e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 4.10e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 8.19e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 7.82e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.74e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 1.70e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.35e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.51e4iT - 8.58e9T^{2} \) |
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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
−13.26180653166741626574972698132, −11.60793100173434117885223557036, −10.66528096935028504956596740118, −10.13495813321060596601848878759, −8.700201151852780137200250543004, −7.50063753504345958763003634895, −4.61724479610303846895160810143, −3.75250175348120823900694112781, −2.43473978369834269816096661792, −0.24949498936858709395961509858,
2.68077955187281505368019847039, 4.91889660570940879340966932811, 6.31689094716702218792124458607, 7.44732646441118717819698399376, 8.343055362313212581052071949299, 9.055537219344741348899582148473, 11.27712857303326435590088780381, 12.70389981888114933457630465017, 13.51264725449046578420832595269, 14.83528443708422850970843759284