L(s) = 1 | + (0.568 + 0.822i)3-s + (−0.748 − 0.663i)4-s + (1.00 − 0.527i)7-s + (−0.354 + 0.935i)9-s + (0.120 − 0.992i)12-s + 13-s + (0.120 + 0.992i)16-s − 0.709·19-s + (1.00 + 0.527i)21-s + (−0.970 − 0.239i)25-s + (−0.970 + 0.239i)27-s + (−1.10 − 0.271i)28-s + (1.45 − 0.358i)31-s + (0.885 − 0.464i)36-s + (−1.71 + 0.423i)37-s + ⋯ |
L(s) = 1 | + (0.568 + 0.822i)3-s + (−0.748 − 0.663i)4-s + (1.00 − 0.527i)7-s + (−0.354 + 0.935i)9-s + (0.120 − 0.992i)12-s + 13-s + (0.120 + 0.992i)16-s − 0.709·19-s + (1.00 + 0.527i)21-s + (−0.970 − 0.239i)25-s + (−0.970 + 0.239i)27-s + (−1.10 − 0.271i)28-s + (1.45 − 0.358i)31-s + (0.885 − 0.464i)36-s + (−1.71 + 0.423i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9757514977\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9757514977\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.568 - 0.822i)T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 5 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 7 | \( 1 + (-1.00 + 0.527i)T + (0.568 - 0.822i)T^{2} \) |
| 11 | \( 1 + (0.748 - 0.663i)T^{2} \) |
| 17 | \( 1 + (-0.568 + 0.822i)T^{2} \) |
| 19 | \( 1 + 0.709T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.748 + 0.663i)T^{2} \) |
| 31 | \( 1 + (-1.45 + 0.358i)T + (0.885 - 0.464i)T^{2} \) |
| 37 | \( 1 + (1.71 - 0.423i)T + (0.885 - 0.464i)T^{2} \) |
| 41 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 43 | \( 1 + (1.71 + 0.423i)T + (0.885 + 0.464i)T^{2} \) |
| 47 | \( 1 + (-0.120 + 0.992i)T^{2} \) |
| 53 | \( 1 + (-0.568 + 0.822i)T^{2} \) |
| 59 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 61 | \( 1 + (1.32 + 0.695i)T + (0.568 + 0.822i)T^{2} \) |
| 67 | \( 1 + (-1.45 + 1.28i)T + (0.120 - 0.992i)T^{2} \) |
| 71 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 73 | \( 1 + (0.0854 + 0.225i)T + (-0.748 + 0.663i)T^{2} \) |
| 79 | \( 1 + (0.850 - 0.753i)T + (0.120 - 0.992i)T^{2} \) |
| 83 | \( 1 + (0.354 + 0.935i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.241 - 1.98i)T + (-0.970 + 0.239i)T^{2} \) |
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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
−10.80438209196802019907763152749, −10.32998071323027854672039554648, −9.432510888571776696279850928136, −8.435573016860440973368035175127, −8.069732727543898072601970698020, −6.42597248050682178716354827897, −5.23168012125239740154015395219, −4.47542255676939500464166491041, −3.62170539929844476000256089837, −1.75282086124197661285480170356,
1.72151642166313286499722722320, 3.13961411403449164315588176809, 4.24525136418783801095203076729, 5.47037790810535737381318739018, 6.65313079928672132616393376007, 7.77700083948033595135860928335, 8.489150418880300317593314317588, 8.807900308384147230436684203622, 10.06246790432902927227444062778, 11.43472419880545319000565834861