L(s) = 1 | + 3-s + 9-s − 4·13-s + 6·17-s − 4·23-s − 5·25-s + 27-s + 6·29-s − 4·37-s − 4·39-s − 10·41-s − 10·43-s − 8·47-s − 7·49-s + 6·51-s + 12·53-s − 6·59-s + 6·61-s − 10·67-s − 4·69-s + 4·71-s + 10·73-s − 5·75-s + 81-s − 14·83-s + 6·87-s − 6·89-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.10·13-s + 1.45·17-s − 0.834·23-s − 25-s + 0.192·27-s + 1.11·29-s − 0.657·37-s − 0.640·39-s − 1.56·41-s − 1.52·43-s − 1.16·47-s − 49-s + 0.840·51-s + 1.64·53-s − 0.781·59-s + 0.768·61-s − 1.22·67-s − 0.481·69-s + 0.474·71-s + 1.17·73-s − 0.577·75-s + 1/9·81-s − 1.53·83-s + 0.643·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.58850368064243075727354630418, −6.91042766114675189501806529577, −6.16730721359156521534088909169, −5.25334592795948146754645377287, −4.75367801879015810295683157435, −3.70713759001138089741986543406, −3.18208693972284923545981238527, −2.23754418710659814623748713170, −1.43258913239847621768055103794, 0,
1.43258913239847621768055103794, 2.23754418710659814623748713170, 3.18208693972284923545981238527, 3.70713759001138089741986543406, 4.75367801879015810295683157435, 5.25334592795948146754645377287, 6.16730721359156521534088909169, 6.91042766114675189501806529577, 7.58850368064243075727354630418