L(s) = 1 | + 3-s − 7-s − 2·9-s + 5·11-s + 3·13-s − 6·19-s − 21-s − 23-s − 5·25-s − 5·27-s − 2·29-s − 4·31-s + 5·33-s + 8·37-s + 3·39-s − 4·41-s + 5·43-s + 47-s − 6·49-s − 6·57-s − 12·59-s − 61-s + 2·63-s + 9·67-s − 69-s − 6·71-s − 10·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.50·11-s + 0.832·13-s − 1.37·19-s − 0.218·21-s − 0.208·23-s − 25-s − 0.962·27-s − 0.371·29-s − 0.718·31-s + 0.870·33-s + 1.31·37-s + 0.480·39-s − 0.624·41-s + 0.762·43-s + 0.145·47-s − 6/7·49-s − 0.794·57-s − 1.56·59-s − 0.128·61-s + 0.251·63-s + 1.09·67-s − 0.120·69-s − 0.712·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.61775716028495856287934743747, −6.63295430984178026366780363106, −6.18660130115353315343455553749, −5.62632193761852197215627061033, −4.34123130773115903853729330202, −3.88166107379104389380005593762, −3.19693918256683649792833646185, −2.22566158849502907791506143663, −1.41508612883361234288768071756, 0,
1.41508612883361234288768071756, 2.22566158849502907791506143663, 3.19693918256683649792833646185, 3.88166107379104389380005593762, 4.34123130773115903853729330202, 5.62632193761852197215627061033, 6.18660130115353315343455553749, 6.63295430984178026366780363106, 7.61775716028495856287934743747