L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 3·11-s − 13-s − 15-s + 5·17-s − 19-s − 21-s − 3·23-s − 4·25-s − 27-s + 5·29-s − 4·31-s + 3·33-s + 35-s − 5·37-s + 39-s − 8·41-s + 43-s + 45-s − 8·47-s + 49-s − 5·51-s + 6·53-s − 3·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s − 0.258·15-s + 1.21·17-s − 0.229·19-s − 0.218·21-s − 0.625·23-s − 4/5·25-s − 0.192·27-s + 0.928·29-s − 0.718·31-s + 0.522·33-s + 0.169·35-s − 0.821·37-s + 0.160·39-s − 1.24·41-s + 0.152·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.700·51-s + 0.824·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.012010791101878922945989446426, −7.26239186217983479827375624086, −6.51364832663627501293822798977, −5.49038864310940250755555663461, −5.37156070429324926996432003024, −4.34347588289563518623478698905, −3.37908547419896643858536952096, −2.32176501698186749232109677114, −1.40782765790296346859043483557, 0,
1.40782765790296346859043483557, 2.32176501698186749232109677114, 3.37908547419896643858536952096, 4.34347588289563518623478698905, 5.37156070429324926996432003024, 5.49038864310940250755555663461, 6.51364832663627501293822798977, 7.26239186217983479827375624086, 8.012010791101878922945989446426