L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + 5-s + (−0.499 + 0.866i)9-s − 11-s + 0.999·12-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.499 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)20-s + 25-s + 0.999·27-s + (−1 + 1.73i)29-s + (0.5 + 0.866i)33-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + 5-s + (−0.499 + 0.866i)9-s − 11-s + 0.999·12-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.499 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)20-s + 25-s + 0.999·27-s + (−1 + 1.73i)29-s + (0.5 + 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9017286897\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9017286897\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.113039009713901438689183647457, −8.598357303782226854378435486374, −7.72863264483160389278303620991, −7.09752147932774029851318586488, −6.21892941143911630894719812732, −5.47656027123747436087920765821, −4.74326881169303380057128796801, −3.49500381135021437934779937940, −2.46467620292673372629862932994, −1.47249227984080015289716550848,
0.70294758872937407203185684283, 2.26883091674488514851389928509, 3.40261795043505058658331050062, 4.57037316011366837436553061345, 5.31847691003586788130934740913, 5.71384383811787228697317398951, 6.36880838177893281941886311643, 7.63336708363880330355617761327, 8.651479263543325687484549024903, 9.391543915944508440803028473134