L(s) = 1 | + (−0.999 + 0.0251i)2-s + (0.998 − 0.0502i)4-s + (−0.984 − 0.175i)5-s + (−0.997 + 0.0753i)8-s + (0.617 + 0.786i)9-s + (0.988 + 0.150i)10-s + (−1.26 + 0.465i)13-s + (0.994 − 0.100i)16-s + (0.669 − 1.75i)17-s + (−0.637 − 0.770i)18-s + (−0.992 − 0.125i)20-s + (0.938 + 0.344i)25-s + (1.25 − 0.497i)26-s + (1.03 − 1.38i)29-s + (−0.992 + 0.125i)32-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.0251i)2-s + (0.998 − 0.0502i)4-s + (−0.984 − 0.175i)5-s + (−0.997 + 0.0753i)8-s + (0.617 + 0.786i)9-s + (0.988 + 0.150i)10-s + (−1.26 + 0.465i)13-s + (0.994 − 0.100i)16-s + (0.669 − 1.75i)17-s + (−0.637 − 0.770i)18-s + (−0.992 − 0.125i)20-s + (0.938 + 0.344i)25-s + (1.25 − 0.497i)26-s + (1.03 − 1.38i)29-s + (−0.992 + 0.125i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6302894504\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6302894504\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.999 - 0.0251i)T \) |
| 5 | \( 1 + (0.984 + 0.175i)T \) |
good | 3 | \( 1 + (-0.617 - 0.786i)T^{2} \) |
| 7 | \( 1 + (-0.728 - 0.684i)T^{2} \) |
| 11 | \( 1 + (-0.356 - 0.934i)T^{2} \) |
| 13 | \( 1 + (1.26 - 0.465i)T + (0.762 - 0.647i)T^{2} \) |
| 17 | \( 1 + (-0.669 + 1.75i)T + (-0.745 - 0.666i)T^{2} \) |
| 19 | \( 1 + (0.556 + 0.830i)T^{2} \) |
| 23 | \( 1 + (0.947 + 0.320i)T^{2} \) |
| 29 | \( 1 + (-1.03 + 1.38i)T + (-0.285 - 0.958i)T^{2} \) |
| 31 | \( 1 + (0.745 + 0.666i)T^{2} \) |
| 37 | \( 1 + (-0.796 - 1.80i)T + (-0.675 + 0.737i)T^{2} \) |
| 41 | \( 1 + (0.751 - 1.49i)T + (-0.597 - 0.801i)T^{2} \) |
| 43 | \( 1 + (0.929 - 0.368i)T^{2} \) |
| 47 | \( 1 + (-0.162 - 0.986i)T^{2} \) |
| 53 | \( 1 + (-1.13 - 0.230i)T + (0.920 + 0.391i)T^{2} \) |
| 59 | \( 1 + (-0.899 + 0.437i)T^{2} \) |
| 61 | \( 1 + (0.123 + 0.246i)T + (-0.597 + 0.801i)T^{2} \) |
| 67 | \( 1 + (-0.793 + 0.607i)T^{2} \) |
| 71 | \( 1 + (-0.448 + 0.893i)T^{2} \) |
| 73 | \( 1 + (-1.02 + 1.71i)T + (-0.470 - 0.882i)T^{2} \) |
| 79 | \( 1 + (-0.617 - 0.786i)T^{2} \) |
| 83 | \( 1 + (-0.938 - 0.344i)T^{2} \) |
| 89 | \( 1 + (-1.46 + 0.621i)T + (0.693 - 0.720i)T^{2} \) |
| 97 | \( 1 + (-1.06 - 1.50i)T + (-0.332 + 0.942i)T^{2} \) |
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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
−9.200669230448652525528265203580, −8.153276424516962239235533651946, −7.70521751303507671288139967413, −7.17729222176902211226198600479, −6.38368868047989684204023854498, −4.99379463759317230862917363192, −4.56108512409063788002851175413, −3.12020193827313108376995288023, −2.34922242274688630642679078770, −0.904749205296848276271922105487,
0.825867177455509318480635427478, 2.19359585309151977560355554164, 3.37465449848357747081849261281, 3.98804867971367435350582984068, 5.28328081193252457253903570190, 6.26446093127224200793645510661, 7.19552136533274731089425880163, 7.43723816270775072019252073479, 8.441822626445383563739017014792, 8.882665337702589464365078282280