L(s) = 1 | − 2i·5-s − 3·25-s − 2i·29-s + 49-s + 2i·53-s + 2·73-s + 2·97-s + 2i·101-s + ⋯ |
L(s) = 1 | − 2i·5-s − 3·25-s − 2i·29-s + 49-s + 2i·53-s + 2·73-s + 2·97-s + 2i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9859431593\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9859431593\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2iT - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 2iT - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 2iT - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 2T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - 2T + T^{2} \) |
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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
−9.515921858426045983602301046016, −9.059740003860128767748979958533, −8.187545809273683691433180738659, −7.63378908382220094757359811056, −6.21381971161003882077582006974, −5.45780422271770771077743201769, −4.60283690933234830142766355764, −3.90962520960734225613550064608, −2.21553977607805232904353463113, −0.918015980458153970151122476360,
2.05433328193680623125449754459, 3.08854663638726921451892984130, 3.77623447827385301035880408703, 5.19873261023162234974249814724, 6.23329559472587249838751815123, 6.91198719731158661318955679234, 7.48568691682640405444696572155, 8.507164368235856257029258774545, 9.596811998769285891963235777661, 10.32904315598308936279450850125