L(s) = 1 | + 14.0i·2-s + (−13.7 + 23.2i)3-s − 132.·4-s + 135. i·5-s + (−325. − 191. i)6-s − 169.·7-s − 952. i·8-s + (−353. − 637. i)9-s − 1.89e3·10-s + 1.42e3i·11-s + (1.80e3 − 3.07e3i)12-s + 1.38e3·13-s − 2.37e3i·14-s + (−3.14e3 − 1.85e3i)15-s + 4.88e3·16-s + 3.51e3i·17-s + ⋯ |
L(s) = 1 | + 1.75i·2-s + (−0.507 + 0.861i)3-s − 2.06·4-s + 1.08i·5-s + (−1.50 − 0.888i)6-s − 0.493·7-s − 1.85i·8-s + (−0.484 − 0.874i)9-s − 1.89·10-s + 1.06i·11-s + (1.04 − 1.77i)12-s + 0.629·13-s − 0.864i·14-s + (−0.932 − 0.549i)15-s + 1.19·16-s + 0.714i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.712956 - 0.194385i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.712956 - 0.194385i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (13.7 - 23.2i)T \) |
| 23 | \( 1 - 2.53e3iT \) |
good | 2 | \( 1 - 14.0iT - 64T^{2} \) |
| 5 | \( 1 - 135. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 169.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.42e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 1.38e3T + 4.82e6T^{2} \) |
| 17 | \( 1 - 3.51e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 695.T + 4.70e7T^{2} \) |
| 29 | \( 1 - 4.68e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.86e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 5.83e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + 8.86e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 5.16e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 7.47e3iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 2.09e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 3.48e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 5.99e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 3.50e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.37e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 8.56e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 6.96e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 5.17e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 2.86e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.83e5T + 8.32e11T^{2} \) |
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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
−14.87515416840083079919972113995, −14.06306891104802090034286203754, −12.54855548405724764893857505727, −10.82162660185940737686509795911, −9.834473153347897028492594717710, −8.622818243881068223057930634015, −7.02390391421777896752250339957, −6.35360807350252956425905612740, −5.06883504478448622654108979311, −3.63885137488645647765680921606,
0.36201909668394874869964687862, 1.20818427209953883965076497065, 2.84928317569868609413577465324, 4.57260170757572372816426383629, 6.04644010091992585640159920409, 8.211913333288177448725081804883, 9.203609825982295217495884228111, 10.57078566545891791418661211631, 11.64605529663994303453881687797, 12.26481748093990341103541093815