L(s) = 1 | − 7-s + 13-s + (0.5 − 0.866i)25-s + (1.5 + 0.866i)31-s + (0.5 + 0.866i)37-s − 1.73i·43-s + 49-s + (0.5 + 0.866i)61-s + (1.5 + 0.866i)67-s + (1 − 1.73i)73-s + (−1.5 + 0.866i)79-s − 91-s + 97-s + (1.5 − 0.866i)103-s + (−0.5 + 0.866i)109-s + ⋯ |
L(s) = 1 | − 7-s + 13-s + (0.5 − 0.866i)25-s + (1.5 + 0.866i)31-s + (0.5 + 0.866i)37-s − 1.73i·43-s + 49-s + (0.5 + 0.866i)61-s + (1.5 + 0.866i)67-s + (1 − 1.73i)73-s + (−1.5 + 0.866i)79-s − 91-s + 97-s + (1.5 − 0.866i)103-s + (−0.5 + 0.866i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.164561033\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.164561033\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 1.73iT - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−8.694900962424661122857533648938, −8.422613933294693839428940217796, −7.26984110398530456153931427144, −6.54354476762319116323897759549, −6.05580078074835682221721177763, −5.06084949260165392597210493948, −4.09290699334847601988498339620, −3.30435502885153488796315971283, −2.44425578812246109941740019186, −0.982103471695430954523385122772,
1.02213249716943992987137170446, 2.45354232716552858738051869983, 3.34652332016447491344779235242, 4.08080041811156436383945455630, 5.10035387925114202314747347158, 6.10733307279266288330178701927, 6.45851359507801734099783236945, 7.41380445650433216744209111051, 8.195841552536736986962084189561, 8.964051542748206565594137672453