L(s) = 1 | + (−0.885 + 0.464i)3-s + (0.354 + 0.935i)4-s + (−0.222 + 0.902i)7-s + (0.568 − 0.822i)9-s + (−0.748 − 0.663i)12-s − 13-s + (−0.748 + 0.663i)16-s + 1.64i·19-s + (−0.222 − 0.902i)21-s + (−0.120 − 0.992i)25-s + (−0.120 + 0.992i)27-s + (−0.922 + 0.112i)28-s + (1.85 + 0.225i)31-s + (0.970 + 0.239i)36-s + (0.475 + 0.0576i)37-s + ⋯ |
L(s) = 1 | + (−0.885 + 0.464i)3-s + (0.354 + 0.935i)4-s + (−0.222 + 0.902i)7-s + (0.568 − 0.822i)9-s + (−0.748 − 0.663i)12-s − 13-s + (−0.748 + 0.663i)16-s + 1.64i·19-s + (−0.222 − 0.902i)21-s + (−0.120 − 0.992i)25-s + (−0.120 + 0.992i)27-s + (−0.922 + 0.112i)28-s + (1.85 + 0.225i)31-s + (0.970 + 0.239i)36-s + (0.475 + 0.0576i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6600435850\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6600435850\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.885 - 0.464i)T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + (-0.354 - 0.935i)T^{2} \) |
| 5 | \( 1 + (0.120 + 0.992i)T^{2} \) |
| 7 | \( 1 + (0.222 - 0.902i)T + (-0.885 - 0.464i)T^{2} \) |
| 11 | \( 1 + (-0.354 + 0.935i)T^{2} \) |
| 17 | \( 1 + (-0.885 - 0.464i)T^{2} \) |
| 19 | \( 1 - 1.64iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.354 + 0.935i)T^{2} \) |
| 31 | \( 1 + (-1.85 - 0.225i)T + (0.970 + 0.239i)T^{2} \) |
| 37 | \( 1 + (-0.475 - 0.0576i)T + (0.970 + 0.239i)T^{2} \) |
| 41 | \( 1 + (0.568 - 0.822i)T^{2} \) |
| 43 | \( 1 + (0.234 + 1.92i)T + (-0.970 + 0.239i)T^{2} \) |
| 47 | \( 1 + (-0.748 - 0.663i)T^{2} \) |
| 53 | \( 1 + (-0.885 - 0.464i)T^{2} \) |
| 59 | \( 1 + (0.120 + 0.992i)T^{2} \) |
| 61 | \( 1 + (0.688 - 0.169i)T + (0.885 - 0.464i)T^{2} \) |
| 67 | \( 1 + (-1.85 - 0.704i)T + (0.748 + 0.663i)T^{2} \) |
| 71 | \( 1 + (0.568 - 0.822i)T^{2} \) |
| 73 | \( 1 + (-1.09 + 0.753i)T + (0.354 - 0.935i)T^{2} \) |
| 79 | \( 1 + (-0.627 + 1.65i)T + (-0.748 - 0.663i)T^{2} \) |
| 83 | \( 1 + (0.568 + 0.822i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.120 + 0.992i)T^{2} \) |
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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
−11.64516869939598764650639713955, −10.46482042145324701060700157143, −9.789218540793685901331312757033, −8.691247056268792387070921434759, −7.78672305889136123586425109153, −6.66635836399058625110976275422, −5.88669455472781977766376540017, −4.75655647608536940554894542405, −3.66679955168328498820081888320, −2.37411744666729249688853089711,
0.944782389261747953360958677341, 2.53655730238305147466752123619, 4.53044390048567912855944843545, 5.22044525972759150737143927980, 6.44741343413880145472575813346, 6.94601561283530896843376856936, 7.84591671227510897167458674935, 9.513550804175378027927992805839, 10.05113585090467166788024401921, 11.06585442723797117773600949542