L(s) = 1 | + 2.82i·2-s + 9.09i·3-s − 8.00·4-s + (−21.5 − 12.6i)5-s − 25.7·6-s + 94.0·7-s − 22.6i·8-s − 1.77·9-s + (35.7 − 60.9i)10-s + 189. i·11-s − 72.7i·12-s + 27.4i·13-s + 266. i·14-s + (115. − 196. i)15-s + 64.0·16-s + 179.·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 1.01i·3-s − 0.500·4-s + (−0.862 − 0.505i)5-s − 0.714·6-s + 1.91·7-s − 0.353i·8-s − 0.0219·9-s + (0.357 − 0.609i)10-s + 1.56i·11-s − 0.505i·12-s + 0.162i·13-s + 1.35i·14-s + (0.511 − 0.871i)15-s + 0.250·16-s + 0.622·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.759792245\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.759792245\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2.82iT \) |
| 5 | \( 1 + (21.5 + 12.6i)T \) |
| 23 | \( 1 + (117. + 515. i)T \) |
good | 3 | \( 1 - 9.09iT - 81T^{2} \) |
| 7 | \( 1 - 94.0T + 2.40e3T^{2} \) |
| 11 | \( 1 - 189. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 27.4iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 179.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 320. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 801.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 714.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.76e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 990.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.57e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 992. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.10e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 4.65e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 3.79e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.64e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.89e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 4.78e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 7.38e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 940.T + 4.74e7T^{2} \) |
| 89 | \( 1 - 1.03e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.38e3T + 8.85e7T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.95222408815551720511924295028, −10.94854087677973346706306293691, −9.952460203725466669583946513754, −8.921129594009533149733782467683, −7.901511057622096917409838542026, −7.35917639910241859511513880926, −5.39453989006226926182533249387, −4.54680486925006982581727990557, −4.10102296891784246588079699132, −1.55392307291192839829557714422,
0.65754865136750693877527339148, 1.74868375731377265202140623193, 3.23080708938310309481212554244, 4.56863492527635364328111630818, 5.89311761628349089121009181860, 7.58821284276877675606749180090, 7.85121757994441798472291813439, 8.934460930162167883294201866186, 10.65271635218657069352600383510, 11.42099421447960604463921238449