L(s) = 1 | + 1.41i·2-s + (−1 − 1.41i)3-s − 2.00·4-s + (2.12 − 0.707i)5-s + (2.00 − 1.41i)6-s + 7-s − 2.82i·8-s + (−1.00 + 2.82i)9-s + (1.00 + 3i)10-s − 4.24·11-s + (2.00 + 2.82i)12-s − 6i·13-s + 1.41i·14-s + (−3.12 − 2.29i)15-s + 4.00·16-s + 4.24·17-s + ⋯ |
L(s) = 1 | + 0.999i·2-s + (−0.577 − 0.816i)3-s − 1.00·4-s + (0.948 − 0.316i)5-s + (0.816 − 0.577i)6-s + 0.377·7-s − 1.00i·8-s + (−0.333 + 0.942i)9-s + (0.316 + 0.948i)10-s − 1.27·11-s + (0.577 + 0.816i)12-s − 1.66i·13-s + 0.377i·14-s + (−0.805 − 0.592i)15-s + 1.00·16-s + 1.02·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01983 - 0.334324i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01983 - 0.334324i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 5 | \( 1 + (-2.12 + 0.707i)T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 7.07iT - 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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
−10.85548826180716822731874985962, −10.27232604482245198757885544974, −9.061367907453524573939195644925, −7.982460779226936813288556093110, −7.49985951007858922781414537621, −6.25643879256575624130339364211, −5.38154317719248833473137974863, −5.02982559733495413126446465533, −2.73054577108713646790079187748, −0.78525186382678588737342378224,
1.72960187515778712577241904296, 3.13868077823481651915369999190, 4.40162955848078837620502499222, 5.33612847101818379007448476507, 6.15233619526466917963157775568, 7.81706070580839776609200741990, 9.037088269681964406689053738249, 9.846238565111668995706145436266, 10.32685703294782079381520688595, 11.16808615504926863183288843047