L(s) = 1 | + 3·5-s + 2·7-s − 6·11-s − 5·13-s + 3·17-s − 2·19-s − 6·23-s + 4·25-s + 3·29-s − 4·31-s + 6·35-s − 5·37-s + 6·41-s + 10·43-s − 3·49-s − 6·53-s − 18·55-s − 12·59-s − 5·61-s − 15·65-s − 2·67-s − 6·71-s − 73-s − 12·77-s − 10·79-s + 9·85-s + 3·89-s + ⋯ |
L(s) = 1 | + 1.34·5-s + 0.755·7-s − 1.80·11-s − 1.38·13-s + 0.727·17-s − 0.458·19-s − 1.25·23-s + 4/5·25-s + 0.557·29-s − 0.718·31-s + 1.01·35-s − 0.821·37-s + 0.937·41-s + 1.52·43-s − 3/7·49-s − 0.824·53-s − 2.42·55-s − 1.56·59-s − 0.640·61-s − 1.86·65-s − 0.244·67-s − 0.712·71-s − 0.117·73-s − 1.36·77-s − 1.12·79-s + 0.976·85-s + 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.70903556366548608336995672505, −7.41344272514235046881371429053, −6.17409621082457093435073018081, −5.63152910747119948119367051464, −5.05081125666270125581203711694, −4.40166910060312716273633741389, −2.94530833543250408814630098907, −2.33139082573100697227118251464, −1.63208959025720136856911384475, 0,
1.63208959025720136856911384475, 2.33139082573100697227118251464, 2.94530833543250408814630098907, 4.40166910060312716273633741389, 5.05081125666270125581203711694, 5.63152910747119948119367051464, 6.17409621082457093435073018081, 7.41344272514235046881371429053, 7.70903556366548608336995672505