L(s) = 1 | + (0.755 − 0.654i)2-s + (0.142 − 0.989i)4-s + (−0.550 − 0.119i)5-s + (−0.540 − 0.841i)8-s + (−0.494 + 0.270i)10-s + (−1.01 − 1.71i)13-s + (−0.959 − 0.281i)16-s + (1.86 − 0.133i)17-s + (−0.196 + 0.527i)20-s + (−0.620 − 0.283i)25-s + (−1.88 − 0.627i)26-s + (−0.834 + 1.20i)29-s + (−0.909 + 0.415i)32-s + (1.32 − 1.32i)34-s + (−0.726 − 1.75i)37-s + ⋯ |
L(s) = 1 | + (0.755 − 0.654i)2-s + (0.142 − 0.989i)4-s + (−0.550 − 0.119i)5-s + (−0.540 − 0.841i)8-s + (−0.494 + 0.270i)10-s + (−1.01 − 1.71i)13-s + (−0.959 − 0.281i)16-s + (1.86 − 0.133i)17-s + (−0.196 + 0.527i)20-s + (−0.620 − 0.283i)25-s + (−1.88 − 0.627i)26-s + (−0.834 + 1.20i)29-s + (−0.909 + 0.415i)32-s + (1.32 − 1.32i)34-s + (−0.726 − 1.75i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3204 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.409225009\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.409225009\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.755 + 0.654i)T \) |
| 3 | \( 1 \) |
| 89 | \( 1 + (0.212 + 0.977i)T \) |
good | 5 | \( 1 + (0.550 + 0.119i)T + (0.909 + 0.415i)T^{2} \) |
| 7 | \( 1 + (-0.936 + 0.349i)T^{2} \) |
| 11 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 13 | \( 1 + (1.01 + 1.71i)T + (-0.479 + 0.877i)T^{2} \) |
| 17 | \( 1 + (-1.86 + 0.133i)T + (0.989 - 0.142i)T^{2} \) |
| 19 | \( 1 + (0.212 - 0.977i)T^{2} \) |
| 23 | \( 1 + (0.977 + 0.212i)T^{2} \) |
| 29 | \( 1 + (0.834 - 1.20i)T + (-0.349 - 0.936i)T^{2} \) |
| 31 | \( 1 + (-0.977 + 0.212i)T^{2} \) |
| 37 | \( 1 + (0.726 + 1.75i)T + (-0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (0.520 - 0.877i)T + (-0.479 - 0.877i)T^{2} \) |
| 43 | \( 1 + (-0.349 + 0.936i)T^{2} \) |
| 47 | \( 1 + (-0.281 + 0.959i)T^{2} \) |
| 53 | \( 1 + (-1.19 + 1.59i)T + (-0.281 - 0.959i)T^{2} \) |
| 59 | \( 1 + (0.877 - 0.479i)T^{2} \) |
| 61 | \( 1 + (1.96 - 0.0702i)T + (0.997 - 0.0713i)T^{2} \) |
| 67 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 71 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 73 | \( 1 + (-0.0801 + 0.273i)T + (-0.841 - 0.540i)T^{2} \) |
| 79 | \( 1 + (0.540 - 0.841i)T^{2} \) |
| 83 | \( 1 + (-0.599 + 0.800i)T^{2} \) |
| 97 | \( 1 + (-1.34 + 0.865i)T + (0.415 - 0.909i)T^{2} \) |
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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
−8.542435616799369440729742543703, −7.58390403808023721496202395499, −7.23791417077083232903224269342, −5.82980482440301415619142714257, −5.45122702298841796482000390319, −4.70336059792710469714697339202, −3.52838531398857878910858219282, −3.20628428400623945589052412949, −2.00925399441721051655476949268, −0.64453699942115242160023626402,
1.86715109834721525426368381190, 2.99414424768356542123151453423, 3.87514047473685317005002209257, 4.47233443674386327424178068996, 5.37962532268916629094811796781, 6.08502450985681238526457965294, 7.02489560031824026860488441478, 7.49653825220804530018037745453, 8.093918694480149752066948511156, 9.062907314050704037386492975842