L(s) = 1 | + 5.65·2-s − 29.9i·3-s + 32.0·4-s − 98.3i·5-s − 169. i·6-s + (195. + 281. i)7-s + 181.·8-s − 168.·9-s − 556. i·10-s − 1.04e3·11-s − 958. i·12-s + 3.66e3i·13-s + (1.10e3 + 1.59e3i)14-s − 2.94e3·15-s + 1.02e3·16-s + 1.85e3i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.10i·3-s + 0.500·4-s − 0.787i·5-s − 0.784i·6-s + (0.570 + 0.821i)7-s + 0.353·8-s − 0.230·9-s − 0.556i·10-s − 0.782·11-s − 0.554i·12-s + 1.66i·13-s + (0.403 + 0.580i)14-s − 0.873·15-s + 0.250·16-s + 0.377i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.82397 - 0.953390i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82397 - 0.953390i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 5.65T \) |
| 7 | \( 1 + (-195. - 281. i)T \) |
good | 3 | \( 1 + 29.9iT - 729T^{2} \) |
| 5 | \( 1 + 98.3iT - 1.56e4T^{2} \) |
| 11 | \( 1 + 1.04e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 3.66e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 1.85e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.49e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.16e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 4.16e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 2.20e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 4.91e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 1.80e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.14e5T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.39e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.24e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 2.71e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 3.38e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 1.06e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 2.16e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + 1.11e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 4.51e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 2.25e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 4.11e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.17e5iT - 8.32e11T^{2} \) |
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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
−18.26584224930938946690351683233, −16.69073178740801040981771543129, −15.16277716787190170424509629038, −13.56451171486002962272060594041, −12.61267282436283628360058488894, −11.47619625570003954078704736690, −8.743131243383779283035843669582, −6.97470933455247246630931941210, −5.04012306530305913138641290254, −1.86371346581523819161058772347,
3.44123114552857990266343436475, 5.21015436235711202081742289833, 7.56705771619632141047465505399, 10.24306775010029967900648671490, 10.94433243691035871113541021821, 13.09515784612772290387638888658, 14.65785795898302620700388099959, 15.41895143814995947830683683374, 16.83510291101240944324770906585, 18.44916164614678172508165051739