L(s) = 1 | + (0.568 − 0.822i)2-s + (−0.354 − 0.935i)4-s + (−0.120 + 0.992i)5-s + (−0.970 − 0.239i)8-s + (−0.568 + 0.822i)9-s + (0.748 + 0.663i)10-s + (0.970 − 0.239i)13-s + (−0.748 + 0.663i)16-s + (−0.447 + 1.81i)17-s + (0.354 + 0.935i)18-s + (0.970 − 0.239i)20-s + (−0.970 − 0.239i)25-s + (0.354 − 0.935i)26-s + (−1.00 + 1.45i)29-s + (0.120 + 0.992i)32-s + ⋯ |
L(s) = 1 | + (0.568 − 0.822i)2-s + (−0.354 − 0.935i)4-s + (−0.120 + 0.992i)5-s + (−0.970 − 0.239i)8-s + (−0.568 + 0.822i)9-s + (0.748 + 0.663i)10-s + (0.970 − 0.239i)13-s + (−0.748 + 0.663i)16-s + (−0.447 + 1.81i)17-s + (0.354 + 0.935i)18-s + (0.970 − 0.239i)20-s + (−0.970 − 0.239i)25-s + (0.354 − 0.935i)26-s + (−1.00 + 1.45i)29-s + (0.120 + 0.992i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.247480380\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.247480380\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.568 + 0.822i)T \) |
| 5 | \( 1 + (0.120 - 0.992i)T \) |
| 13 | \( 1 + (-0.970 + 0.239i)T \) |
good | 3 | \( 1 + (0.568 - 0.822i)T^{2} \) |
| 7 | \( 1 + (-0.885 - 0.464i)T^{2} \) |
| 11 | \( 1 + (-0.354 + 0.935i)T^{2} \) |
| 17 | \( 1 + (0.447 - 1.81i)T + (-0.885 - 0.464i)T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (1.00 - 1.45i)T + (-0.354 - 0.935i)T^{2} \) |
| 31 | \( 1 + (-0.970 - 0.239i)T^{2} \) |
| 37 | \( 1 + (0.180 - 1.48i)T + (-0.970 - 0.239i)T^{2} \) |
| 41 | \( 1 + (0.764 + 1.45i)T + (-0.568 + 0.822i)T^{2} \) |
| 43 | \( 1 + (-0.970 + 0.239i)T^{2} \) |
| 47 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 53 | \( 1 + (-0.114 + 0.464i)T + (-0.885 - 0.464i)T^{2} \) |
| 59 | \( 1 + (0.120 + 0.992i)T^{2} \) |
| 61 | \( 1 + (-1.71 + 0.423i)T + (0.885 - 0.464i)T^{2} \) |
| 67 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 71 | \( 1 + (0.568 - 0.822i)T^{2} \) |
| 73 | \( 1 + (-1.13 - 1.64i)T + (-0.354 + 0.935i)T^{2} \) |
| 79 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 83 | \( 1 + (-0.568 - 0.822i)T^{2} \) |
| 89 | \( 1 + 1.32iT - T^{2} \) |
| 97 | \( 1 + (-0.180 + 0.159i)T + (0.120 - 0.992i)T^{2} \) |
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
−8.671761628755926364926100878992, −8.442720162707367486004704035644, −7.25021900264641311146029518016, −6.41018064821124004151177482336, −5.78695767289563452745425589359, −5.05538068349408712644075947057, −3.83715357888601553130770947562, −3.48127067683739416500989693439, −2.40732949623937286026294246629, −1.62069682164278821922300952750,
0.60449211509219594682825792495, 2.37033550112154686413848818541, 3.52685860719350768839987813033, 4.14302979831630649685704993414, 5.02405820467387910073980359412, 5.69081191151060383197395229130, 6.36120271189793943934320107738, 7.19190862463913193104873312701, 7.950575407567353272138812314963, 8.703134236691893232671205896107