L(s) = 1 | + (−0.980 − 0.198i)3-s + (−0.270 − 0.962i)4-s + (0.542 + 0.840i)7-s + (0.921 + 0.388i)9-s + (0.0747 + 0.997i)12-s + (0.636 − 0.0317i)13-s + (−0.853 + 0.521i)16-s + (0.222 + 0.974i)19-s + (−0.365 − 0.930i)21-s + (0.124 + 0.992i)25-s + (−0.826 − 0.563i)27-s + (0.661 − 0.749i)28-s − 1.93·31-s + (0.124 − 0.992i)36-s + (1.08 + 1.59i)37-s + ⋯ |
L(s) = 1 | + (−0.980 − 0.198i)3-s + (−0.270 − 0.962i)4-s + (0.542 + 0.840i)7-s + (0.921 + 0.388i)9-s + (0.0747 + 0.997i)12-s + (0.636 − 0.0317i)13-s + (−0.853 + 0.521i)16-s + (0.222 + 0.974i)19-s + (−0.365 − 0.930i)21-s + (0.124 + 0.992i)25-s + (−0.826 − 0.563i)27-s + (0.661 − 0.749i)28-s − 1.93·31-s + (0.124 − 0.992i)36-s + (1.08 + 1.59i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8339362291\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8339362291\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.980 + 0.198i)T \) |
| 7 | \( 1 + (-0.542 - 0.840i)T \) |
| 19 | \( 1 + (-0.222 - 0.974i)T \) |
good | 2 | \( 1 + (0.270 + 0.962i)T^{2} \) |
| 5 | \( 1 + (-0.124 - 0.992i)T^{2} \) |
| 11 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (-0.636 + 0.0317i)T + (0.995 - 0.0995i)T^{2} \) |
| 17 | \( 1 + (-0.853 - 0.521i)T^{2} \) |
| 23 | \( 1 + (0.0249 + 0.999i)T^{2} \) |
| 29 | \( 1 + (0.980 + 0.198i)T^{2} \) |
| 31 | \( 1 + 1.93T + T^{2} \) |
| 37 | \( 1 + (-1.08 - 1.59i)T + (-0.365 + 0.930i)T^{2} \) |
| 41 | \( 1 + (0.124 + 0.992i)T^{2} \) |
| 43 | \( 1 + (-0.0459 - 1.84i)T + (-0.998 + 0.0498i)T^{2} \) |
| 47 | \( 1 + (0.995 - 0.0995i)T^{2} \) |
| 53 | \( 1 + (-0.853 + 0.521i)T^{2} \) |
| 59 | \( 1 + (-0.542 - 0.840i)T^{2} \) |
| 61 | \( 1 + (-0.0491 + 0.491i)T + (-0.980 - 0.198i)T^{2} \) |
| 67 | \( 1 + (-1.21 + 1.45i)T + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (-0.661 - 0.749i)T^{2} \) |
| 73 | \( 1 + (-1.27 + 0.653i)T + (0.583 - 0.811i)T^{2} \) |
| 79 | \( 1 + (-0.623 + 1.71i)T + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (0.0747 + 0.997i)T^{2} \) |
| 89 | \( 1 + (-0.270 + 0.962i)T^{2} \) |
| 97 | \( 1 + (1.37 + 0.501i)T + (0.766 + 0.642i)T^{2} \) |
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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
−9.259383982642252335594656117193, −8.235376078627687899570051002470, −7.51905691990990773666212728411, −6.34915479228192493032173901219, −6.01591521895240884979290455733, −5.20371070758442315430413488461, −4.72904419198952320915193149711, −3.54013988646514979016592668277, −1.95500006417666435964194268237, −1.24868188878037167382302090498,
0.70132182927891253542825197811, 2.24310572280930145318567243242, 3.73694432512212714613567355278, 4.08261515862209721595071824226, 4.99877620191415591400316710017, 5.75233789968669131798530676574, 6.97194927915596395605240174877, 7.18232206404451035524621949586, 8.142160870330001531121584681027, 8.920563611207270525620761141383