L(s) = 1 | − 4.30·3-s + 7.29i·5-s − 8.43·9-s − 21.8i·11-s + 30.5i·13-s − 31.4i·15-s − 55.0i·17-s − 58.7·19-s − 79.6i·23-s + 71.7·25-s + 152.·27-s − 116.·29-s + 110.·31-s + 94.0i·33-s − 376.·37-s + ⋯ |
L(s) = 1 | − 0.829·3-s + 0.652i·5-s − 0.312·9-s − 0.598i·11-s + 0.651i·13-s − 0.541i·15-s − 0.786i·17-s − 0.708·19-s − 0.722i·23-s + 0.574·25-s + 1.08·27-s − 0.744·29-s + 0.637·31-s + 0.496i·33-s − 1.67·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.075540153\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.075540153\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 4.30T + 27T^{2} \) |
| 5 | \( 1 - 7.29iT - 125T^{2} \) |
| 11 | \( 1 + 21.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 30.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 55.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 58.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 79.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 116.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 376.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 232. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 260. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 36.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 59.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 707.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 323. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 239. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 887. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 550. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.24e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.39e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 982. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 441. iT - 9.12e5T^{2} \) |
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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
−10.24093661567891628976893006491, −9.043806727294480008818042794389, −8.344368616236539004399309084531, −7.02569653486332502212722593176, −6.51828598275922435048589609754, −5.58443035629332967628630002941, −4.68001010865769391089094790398, −3.41215617634945988188027367925, −2.31636908643622150086100714731, −0.62257089253553375019591919349,
0.57081937387088636796051601035, 1.90464019794720350257078467147, 3.42874811250008645993831007537, 4.65611021134235539496419527962, 5.38827916371006621312988611189, 6.18271721392738965931187040872, 7.17156913669638986588618799503, 8.266034814941413741482565586715, 8.904824687472099710151864195629, 10.02212993577254528750183788132