L(s) = 1 | + (0.120 + 0.992i)3-s + (0.885 − 0.464i)4-s + (−0.180 + 0.159i)7-s + (−0.970 + 0.239i)9-s + (0.568 + 0.822i)12-s + 13-s + (0.568 − 0.822i)16-s − 1.94·19-s + (−0.180 − 0.159i)21-s + (−0.354 + 0.935i)25-s + (−0.354 − 0.935i)27-s + (−0.0854 + 0.225i)28-s + (−0.627 − 1.65i)31-s + (−0.748 + 0.663i)36-s + (0.530 + 1.39i)37-s + ⋯ |
L(s) = 1 | + (0.120 + 0.992i)3-s + (0.885 − 0.464i)4-s + (−0.180 + 0.159i)7-s + (−0.970 + 0.239i)9-s + (0.568 + 0.822i)12-s + 13-s + (0.568 − 0.822i)16-s − 1.94·19-s + (−0.180 − 0.159i)21-s + (−0.354 + 0.935i)25-s + (−0.354 − 0.935i)27-s + (−0.0854 + 0.225i)28-s + (−0.627 − 1.65i)31-s + (−0.748 + 0.663i)36-s + (0.530 + 1.39i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.043743021\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.043743021\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.120 - 0.992i)T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 5 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 7 | \( 1 + (0.180 - 0.159i)T + (0.120 - 0.992i)T^{2} \) |
| 11 | \( 1 + (-0.885 - 0.464i)T^{2} \) |
| 17 | \( 1 + (-0.120 + 0.992i)T^{2} \) |
| 19 | \( 1 + 1.94T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 31 | \( 1 + (0.627 + 1.65i)T + (-0.748 + 0.663i)T^{2} \) |
| 37 | \( 1 + (-0.530 - 1.39i)T + (-0.748 + 0.663i)T^{2} \) |
| 41 | \( 1 + (0.970 - 0.239i)T^{2} \) |
| 43 | \( 1 + (-0.530 + 1.39i)T + (-0.748 - 0.663i)T^{2} \) |
| 47 | \( 1 + (-0.568 - 0.822i)T^{2} \) |
| 53 | \( 1 + (-0.120 + 0.992i)T^{2} \) |
| 59 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 61 | \( 1 + (1.32 + 1.17i)T + (0.120 + 0.992i)T^{2} \) |
| 67 | \( 1 + (0.627 + 0.329i)T + (0.568 + 0.822i)T^{2} \) |
| 71 | \( 1 + (0.970 - 0.239i)T^{2} \) |
| 73 | \( 1 + (1.10 + 0.271i)T + (0.885 + 0.464i)T^{2} \) |
| 79 | \( 1 + (-0.213 - 0.112i)T + (0.568 + 0.822i)T^{2} \) |
| 83 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1.13 + 1.64i)T + (-0.354 - 0.935i)T^{2} \) |
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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
−11.01161050751841250293800568786, −10.49772023441173479507436869239, −9.539377620579378409183663370791, −8.711142299328751114411223960387, −7.68083202984683750479827736526, −6.30748362377606855223624088807, −5.80940327356968749208548626921, −4.48368264033672289791408752688, −3.37949326430949326603062545325, −2.09987512377027086902394265016,
1.73013882416810019294517259231, 2.85670948357188494551015663799, 4.06239127593891400489189079797, 5.92993631640439693871875434226, 6.49144299309225133249445250382, 7.34767911279986896026955672839, 8.275249664538902716456163107870, 8.888624309055078522451229209575, 10.52533898451594106783145202185, 11.05021872200800012082841948694