L(s) = 1 | + (0.190 + 1.81i)5-s + (−0.809 + 0.587i)9-s + (0.809 − 1.40i)13-s + (−0.773 + 1.73i)17-s + (−2.28 + 0.486i)25-s + (−0.360 + 0.207i)29-s + (1.64 + 0.535i)37-s + (−0.0646 + 0.198i)41-s + (−1.22 − 1.35i)45-s + (0.104 − 0.994i)49-s + (1.16 − 1.60i)53-s + (0.978 + 0.207i)61-s + (2.70 + 1.20i)65-s + (−0.104 + 0.994i)73-s + (0.309 − 0.951i)81-s + ⋯ |
L(s) = 1 | + (0.190 + 1.81i)5-s + (−0.809 + 0.587i)9-s + (0.809 − 1.40i)13-s + (−0.773 + 1.73i)17-s + (−2.28 + 0.486i)25-s + (−0.360 + 0.207i)29-s + (1.64 + 0.535i)37-s + (−0.0646 + 0.198i)41-s + (−1.22 − 1.35i)45-s + (0.104 − 0.994i)49-s + (1.16 − 1.60i)53-s + (0.978 + 0.207i)61-s + (2.70 + 1.20i)65-s + (−0.104 + 0.994i)73-s + (0.309 − 0.951i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0733 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0733 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9529422162\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9529422162\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 61 | \( 1 + (-0.978 - 0.207i)T \) |
good | 3 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.190 - 1.81i)T + (-0.978 + 0.207i)T^{2} \) |
| 7 | \( 1 + (-0.104 + 0.994i)T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.773 - 1.73i)T + (-0.669 - 0.743i)T^{2} \) |
| 19 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 23 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 29 | \( 1 + (0.360 - 0.207i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.913 - 0.406i)T^{2} \) |
| 37 | \( 1 + (-1.64 - 0.535i)T + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.0646 - 0.198i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + (0.669 - 0.743i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.16 + 1.60i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.913 + 0.406i)T^{2} \) |
| 67 | \( 1 + (-0.978 + 0.207i)T^{2} \) |
| 71 | \( 1 + (-0.978 - 0.207i)T^{2} \) |
| 73 | \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
| 79 | \( 1 + (0.669 - 0.743i)T^{2} \) |
| 83 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 89 | \( 1 + (-1.41 + 0.459i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.204 - 0.0434i)T + (0.913 - 0.406i)T^{2} \) |
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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
−10.55232425563868950231737007239, −9.901572465850611587665798875193, −8.467682855621372888378388847675, −8.012610563281401540879668291167, −6.92566209174327966598933552042, −6.12780129073240763884938124462, −5.54931957653664785316511917738, −3.89444750716894409893765904875, −3.06435835141547125185599767397, −2.14803224756157399886767097209,
0.945163378470043606268693501130, 2.37967187798592542264555704070, 3.98840147936386232064070534579, 4.69783866881740784903046954753, 5.63427850864588933492832745746, 6.44730022880872827867575963599, 7.62637358902958105821193860401, 8.681831037028280215064929140291, 9.177856837653697814390241492919, 9.486803733168466713126048417708