L(s) = 1 | + 2·3-s + 7-s + 9-s − 3·11-s − 13-s + 19-s + 2·21-s − 23-s − 4·27-s − 3·29-s − 2·31-s − 6·33-s + 2·37-s − 2·39-s + 3·41-s + 43-s − 6·49-s − 12·53-s + 2·57-s + 6·59-s + 2·61-s + 63-s − 8·67-s − 2·69-s + 6·71-s − 7·73-s − 3·77-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.229·19-s + 0.436·21-s − 0.208·23-s − 0.769·27-s − 0.557·29-s − 0.359·31-s − 1.04·33-s + 0.328·37-s − 0.320·39-s + 0.468·41-s + 0.152·43-s − 6/7·49-s − 1.64·53-s + 0.264·57-s + 0.781·59-s + 0.256·61-s + 0.125·63-s − 0.977·67-s − 0.240·69-s + 0.712·71-s − 0.819·73-s − 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.65002721082632609200722036671, −6.91455280588508681626939281594, −5.92797838968929404789038515173, −5.28697834690244400556048162104, −4.51368047706232537231115928321, −3.69746724152175177080603873962, −2.94060991841787652180730775014, −2.34746491892901115419717792287, −1.50001760183063569158676053738, 0,
1.50001760183063569158676053738, 2.34746491892901115419717792287, 2.94060991841787652180730775014, 3.69746724152175177080603873962, 4.51368047706232537231115928321, 5.28697834690244400556048162104, 5.92797838968929404789038515173, 6.91455280588508681626939281594, 7.65002721082632609200722036671