L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s + 5·13-s − 6·17-s + 6·19-s + 2·21-s − 9·23-s − 27-s − 6·29-s − 5·31-s + 33-s − 6·37-s − 5·39-s + 3·41-s − 43-s + 13·47-s − 3·49-s + 6·51-s − 13·53-s − 6·57-s + 9·59-s + 8·61-s − 2·63-s + 4·67-s + 9·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.38·13-s − 1.45·17-s + 1.37·19-s + 0.436·21-s − 1.87·23-s − 0.192·27-s − 1.11·29-s − 0.898·31-s + 0.174·33-s − 0.986·37-s − 0.800·39-s + 0.468·41-s − 0.152·43-s + 1.89·47-s − 3/7·49-s + 0.840·51-s − 1.78·53-s − 0.794·57-s + 1.17·59-s + 1.02·61-s − 0.251·63-s + 0.488·67-s + 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 15 T + p T^{2} \) |
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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
−14.75468411487212, −13.98036147504503, −13.73853969470832, −13.20256300847098, −12.67824715149486, −12.26818736985898, −11.56658581059893, −11.00169513182102, −10.92013113146099, −10.00639596962838, −9.642903354111107, −9.101375428003831, −8.489333898264812, −7.909496140879211, −7.243383727696936, −6.732635857726032, −6.160044408741046, −5.704944470589384, −5.222146420098614, −4.356207913861246, −3.680558110734283, −3.463641900911470, −2.310538813221048, −1.804835666283428, −0.7876595712263313, 0,
0.7876595712263313, 1.804835666283428, 2.310538813221048, 3.463641900911470, 3.680558110734283, 4.356207913861246, 5.222146420098614, 5.704944470589384, 6.160044408741046, 6.732635857726032, 7.243383727696936, 7.909496140879211, 8.489333898264812, 9.101375428003831, 9.642903354111107, 10.00639596962838, 10.92013113146099, 11.00169513182102, 11.56658581059893, 12.26818736985898, 12.67824715149486, 13.20256300847098, 13.73853969470832, 13.98036147504503, 14.75468411487212