L(s) = 1 | + (1.43 − 1.43i)2-s + 3-s − 2.12i·4-s + (−2.46 − 2.46i)5-s + (1.43 − 1.43i)6-s + (−0.806 − 0.806i)7-s + (−0.184 − 0.184i)8-s + 9-s − 7.09·10-s + (−3.27 − 0.500i)11-s − 2.12i·12-s + (0.994 − 3.46i)13-s − 2.31·14-s + (−2.46 − 2.46i)15-s + 3.72·16-s + 1.84·17-s + ⋯ |
L(s) = 1 | + (1.01 − 1.01i)2-s + 0.577·3-s − 1.06i·4-s + (−1.10 − 1.10i)5-s + (0.586 − 0.586i)6-s + (−0.304 − 0.304i)7-s + (−0.0653 − 0.0653i)8-s + 0.333·9-s − 2.24·10-s + (−0.988 − 0.150i)11-s − 0.614i·12-s + (0.275 − 0.961i)13-s − 0.619·14-s + (−0.637 − 0.637i)15-s + 0.931·16-s + 0.448·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.927735 - 2.06145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.927735 - 2.06145i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 11 | \( 1 + (3.27 + 0.500i)T \) |
| 13 | \( 1 + (-0.994 + 3.46i)T \) |
good | 2 | \( 1 + (-1.43 + 1.43i)T - 2iT^{2} \) |
| 5 | \( 1 + (2.46 + 2.46i)T + 5iT^{2} \) |
| 7 | \( 1 + (0.806 + 0.806i)T + 7iT^{2} \) |
| 17 | \( 1 - 1.84T + 17T^{2} \) |
| 19 | \( 1 + (-5.23 + 5.23i)T - 19iT^{2} \) |
| 23 | \( 1 - 8.45iT - 23T^{2} \) |
| 29 | \( 1 - 5.73iT - 29T^{2} \) |
| 31 | \( 1 + (-4.19 - 4.19i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.19 + 1.19i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.04 + 1.04i)T - 41iT^{2} \) |
| 43 | \( 1 + 2.53T + 43T^{2} \) |
| 47 | \( 1 + (2.58 - 2.58i)T - 47iT^{2} \) |
| 53 | \( 1 - 3.52T + 53T^{2} \) |
| 59 | \( 1 + (8.26 - 8.26i)T - 59iT^{2} \) |
| 61 | \( 1 + 2.26iT - 61T^{2} \) |
| 67 | \( 1 + (10.7 + 10.7i)T + 67iT^{2} \) |
| 71 | \( 1 + (-3.91 - 3.91i)T + 71iT^{2} \) |
| 73 | \( 1 + (-7.89 - 7.89i)T + 73iT^{2} \) |
| 79 | \( 1 + 3.25iT - 79T^{2} \) |
| 83 | \( 1 + (-2.19 + 2.19i)T - 83iT^{2} \) |
| 89 | \( 1 + (-11.1 + 11.1i)T - 89iT^{2} \) |
| 97 | \( 1 + (-2.64 - 2.64i)T + 97iT^{2} \) |
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
−11.09324953561721364972220408876, −10.18675335088000005007710301006, −9.081778899775985340670525834672, −8.000125718167952625190305503666, −7.46017018955900653525393608050, −5.37896959119234243074507211129, −4.83625090045129899503687172392, −3.56148066135517694640519896006, −3.04156986390671410179363370650, −1.09419394800807271180770569965,
2.75232333853191709821040854835, 3.71716572032666832151700052948, 4.60313458252558887748922769022, 5.99825112251811516765750191778, 6.78138145429596249116738803517, 7.73307619202250647511324164881, 8.126127409865762187526773690612, 9.743356285714476411076358984724, 10.58658054922672095592686677468, 11.79111824372385247089604581332