L(s) = 1 | + (−0.654 + 1.25i)2-s + i·3-s + (−1.14 − 1.64i)4-s + (−0.206 − 2.22i)5-s + (−1.25 − 0.654i)6-s + (−0.859 + 2.50i)7-s + (2.80 − 0.359i)8-s − 9-s + (2.92 + 1.19i)10-s + 4.14i·11-s + (1.64 − 1.14i)12-s − 3.35·13-s + (−2.57 − 2.71i)14-s + (2.22 − 0.206i)15-s + (−1.38 + 3.75i)16-s − 3.18·17-s + ⋯ |
L(s) = 1 | + (−0.462 + 0.886i)2-s + 0.577i·3-s + (−0.571 − 0.820i)4-s + (−0.0925 − 0.995i)5-s + (−0.511 − 0.267i)6-s + (−0.324 + 0.945i)7-s + (0.991 − 0.127i)8-s − 0.333·9-s + (0.925 + 0.378i)10-s + 1.24i·11-s + (0.473 − 0.330i)12-s − 0.930·13-s + (−0.688 − 0.725i)14-s + (0.574 − 0.0534i)15-s + (−0.346 + 0.938i)16-s − 0.773·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0750080 - 0.400375i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0750080 - 0.400375i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.654 - 1.25i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.206 + 2.22i)T \) |
| 7 | \( 1 + (0.859 - 2.50i)T \) |
good | 11 | \( 1 - 4.14iT - 11T^{2} \) |
| 13 | \( 1 + 3.35T + 13T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 + 6.52T + 19T^{2} \) |
| 23 | \( 1 + 1.42T + 23T^{2} \) |
| 29 | \( 1 - 8.88T + 29T^{2} \) |
| 31 | \( 1 + 2.93T + 31T^{2} \) |
| 37 | \( 1 - 9.54iT - 37T^{2} \) |
| 41 | \( 1 + 3.55iT - 41T^{2} \) |
| 43 | \( 1 + 0.667T + 43T^{2} \) |
| 47 | \( 1 - 0.693iT - 47T^{2} \) |
| 53 | \( 1 + 4.18iT - 53T^{2} \) |
| 59 | \( 1 - 5.96T + 59T^{2} \) |
| 61 | \( 1 - 10.3iT - 61T^{2} \) |
| 67 | \( 1 + 7.05T + 67T^{2} \) |
| 71 | \( 1 - 9.50iT - 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 - 5.25iT - 83T^{2} \) |
| 89 | \( 1 - 2.83iT - 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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
−11.80338006337793043310969958611, −10.30874584004585320377914631598, −9.740704721192616044094278019255, −8.835982978023929023602470813664, −8.352487615136685924587605426768, −7.04877926148439919062701167306, −6.03670664313479275350660745633, −4.88650349346244434886337533180, −4.42656031603292662614423957588, −2.14790509587705965079946337554,
0.28729502575631832677251605933, 2.24557944017850127467698416043, 3.28898928482653963709076482414, 4.39009357749239086512751834979, 6.24149367711401677082618241897, 7.09149004745221730407025206984, 7.966786677918418398943301078030, 8.896666091518104969650116586135, 10.10905101285248847603274696489, 10.75424865404157566172245803875