L(s) = 1 | + 2.82i·2-s − 6.81i·3-s − 8.00·4-s + (−6.15 + 24.2i)5-s + 19.2·6-s − 68.2·7-s − 22.6i·8-s + 34.5·9-s + (−68.5 − 17.4i)10-s − 137. i·11-s + 54.5i·12-s − 37.8i·13-s − 193. i·14-s + (165. + 41.9i)15-s + 64.0·16-s + 205.·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.757i·3-s − 0.500·4-s + (−0.246 + 0.969i)5-s + 0.535·6-s − 1.39·7-s − 0.353i·8-s + 0.426·9-s + (−0.685 − 0.174i)10-s − 1.13i·11-s + 0.378i·12-s − 0.223i·13-s − 0.984i·14-s + (0.733 + 0.186i)15-s + 0.250·16-s + 0.709·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.397677694\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.397677694\) |
\(L(3)\) |
|
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 | \( 1 + (6.15 - 24.2i)T \) |
| 23 | \( 1 + (484. - 212. i)T \) |
good | 3 | \( 1 + 6.81iT - 81T^{2} \) |
| 7 | \( 1 + 68.2T + 2.40e3T^{2} \) |
| 11 | \( 1 + 137. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 37.8iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 205.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 403. iT - 1.30e5T^{2} \) |
| 29 | \( 1 - 929.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.63e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.55e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.97e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 394.T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.90e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.40e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 4.24e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 630. iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 492.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.92e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 6.58e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 739. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 6.24e3T + 4.74e7T^{2} \) |
| 89 | \( 1 - 9.50e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 3.19e3T + 8.85e7T^{2} \) |
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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
−11.81316061431404240291174283806, −10.32619490035066860844733323626, −9.756873439192574547286284712118, −8.185065196939939251880296375260, −7.51141142561448123211414304546, −6.27619032223017201499319901167, −6.11650133444903089279746595289, −3.91717837620940973907445906644, −2.85935417028631081996139258753, −0.69388265668926619779038596987,
0.879125358281766103157983092139, 2.73244287371595994437206314623, 4.17348542494858074872661229469, 4.69776335952222094931065687652, 6.27942972333670677973325527101, 7.69344338735468495302997019010, 9.075341576069314505883893198425, 9.732863164217747635663045223340, 10.20726852936605374655987372017, 11.62509145687958387535550472838