L(s) = 1 | + (−0.0623 − 1.30i)2-s + (−0.712 + 0.0680i)4-s + (−0.888 − 0.458i)7-s + (−0.0530 − 0.368i)8-s + (0.841 − 0.540i)9-s + (−0.0311 + 0.0899i)11-s + (−0.544 + 1.19i)14-s + (−1.18 + 0.227i)16-s + (−0.759 − 1.06i)18-s + (0.119 + 0.0351i)22-s + (0.437 − 1.80i)23-s + (−0.142 + 0.989i)25-s + (0.664 + 0.265i)28-s + (0.888 + 1.53i)29-s + (0.283 + 1.16i)32-s + ⋯ |
L(s) = 1 | + (−0.0623 − 1.30i)2-s + (−0.712 + 0.0680i)4-s + (−0.888 − 0.458i)7-s + (−0.0530 − 0.368i)8-s + (0.841 − 0.540i)9-s + (−0.0311 + 0.0899i)11-s + (−0.544 + 1.19i)14-s + (−1.18 + 0.227i)16-s + (−0.759 − 1.06i)18-s + (0.119 + 0.0351i)22-s + (0.437 − 1.80i)23-s + (−0.142 + 0.989i)25-s + (0.664 + 0.265i)28-s + (0.888 + 1.53i)29-s + (0.283 + 1.16i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8173838299\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8173838299\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.888 + 0.458i)T \) |
| 67 | \( 1 + (-0.981 + 0.189i)T \) |
good | 2 | \( 1 + (0.0623 + 1.30i)T + (-0.995 + 0.0950i)T^{2} \) |
| 3 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 5 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 11 | \( 1 + (0.0311 - 0.0899i)T + (-0.786 - 0.618i)T^{2} \) |
| 13 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 17 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 19 | \( 1 + (-0.580 + 0.814i)T^{2} \) |
| 23 | \( 1 + (-0.437 + 1.80i)T + (-0.888 - 0.458i)T^{2} \) |
| 29 | \( 1 + (-0.888 - 1.53i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.235 + 0.971i)T^{2} \) |
| 37 | \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.327 + 0.945i)T^{2} \) |
| 43 | \( 1 + (-0.481 - 1.05i)T + (-0.654 + 0.755i)T^{2} \) |
| 47 | \( 1 + (-0.0475 - 0.998i)T^{2} \) |
| 53 | \( 1 + (-0.601 + 1.31i)T + (-0.654 - 0.755i)T^{2} \) |
| 59 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 61 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 71 | \( 1 + (-0.651 + 0.0621i)T + (0.981 - 0.189i)T^{2} \) |
| 73 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 79 | \( 1 + (1.78 - 0.713i)T + (0.723 - 0.690i)T^{2} \) |
| 83 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 89 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−10.78054022002447124229046204259, −10.18283110363292231257794229406, −9.537525405485877624461480756264, −8.590762472243069334154857117180, −6.97544814880599414984410135809, −6.57807461865206546044282207721, −4.77211238923418245872915908650, −3.71749394962174599877192736616, −2.85021183903386048119490560132, −1.23183718474848882676955753624,
2.38347050101406084652669767212, 4.01680774713635175913240823422, 5.30723661739913326888430940164, 6.05751461528859527879943925977, 7.03511347843433993028314580043, 7.67375847208286602996436211503, 8.706858754246101860154019155610, 9.559539411082526685298462618743, 10.45958626491124554678389437160, 11.63584715347201618923052112698