L(s) = 1 | + 3-s − 2·7-s − 2·9-s − 4·11-s + 5·13-s + 2·17-s + 6·19-s − 2·21-s − 23-s − 5·27-s + 29-s − 9·31-s − 4·33-s + 4·37-s + 5·39-s + 3·41-s − 8·43-s + 5·47-s − 3·49-s + 2·51-s − 6·53-s + 6·57-s − 4·59-s − 10·61-s + 4·63-s + 4·67-s − 69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.755·7-s − 2/3·9-s − 1.20·11-s + 1.38·13-s + 0.485·17-s + 1.37·19-s − 0.436·21-s − 0.208·23-s − 0.962·27-s + 0.185·29-s − 1.61·31-s − 0.696·33-s + 0.657·37-s + 0.800·39-s + 0.468·41-s − 1.21·43-s + 0.729·47-s − 3/7·49-s + 0.280·51-s − 0.824·53-s + 0.794·57-s − 0.520·59-s − 1.28·61-s + 0.503·63-s + 0.488·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−7.969052501649846613687597108527, −7.45863842006705792194430065803, −6.43734707501955635173408721604, −5.70627731025313504370479857712, −5.21348643867460234828258841434, −3.89283956394425282317299717155, −3.23125986358985050022207661843, −2.70734930373685116678091671614, −1.43557133490489981416489203760, 0,
1.43557133490489981416489203760, 2.70734930373685116678091671614, 3.23125986358985050022207661843, 3.89283956394425282317299717155, 5.21348643867460234828258841434, 5.70627731025313504370479857712, 6.43734707501955635173408721604, 7.45863842006705792194430065803, 7.969052501649846613687597108527