| L(s) = 1 | + 3-s + 5-s + 5·7-s + 9-s − 2·11-s − 5·13-s + 15-s − 2·17-s − 5·19-s + 5·21-s + 25-s + 27-s + 9·29-s + 4·31-s − 2·33-s + 5·35-s − 5·39-s − 8·41-s + 4·43-s + 45-s − 2·47-s + 18·49-s − 2·51-s − 8·53-s − 2·55-s − 5·57-s − 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.88·7-s + 1/3·9-s − 0.603·11-s − 1.38·13-s + 0.258·15-s − 0.485·17-s − 1.14·19-s + 1.09·21-s + 1/5·25-s + 0.192·27-s + 1.67·29-s + 0.718·31-s − 0.348·33-s + 0.845·35-s − 0.800·39-s − 1.24·41-s + 0.609·43-s + 0.149·45-s − 0.291·47-s + 18/7·49-s − 0.280·51-s − 1.09·53-s − 0.269·55-s − 0.662·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.660123871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.660123871\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−12.55727179087527, −12.14702105344254, −11.79990814961824, −11.13836686147809, −10.73677793303801, −10.34878345091586, −9.973280916590532, −9.353694097974730, −8.810870422495194, −8.430753378082330, −8.033768032792740, −7.675499356935150, −7.159357425304693, −6.572012297769611, −6.117761815137711, −5.351137689295675, −4.879617732141908, −4.565651165351640, −4.352446211309648, −3.332245804503982, −2.695849464008673, −2.328053414673685, −1.838230554474560, −1.339326255141184, −0.4548361092035724,
0.4548361092035724, 1.339326255141184, 1.838230554474560, 2.328053414673685, 2.695849464008673, 3.332245804503982, 4.352446211309648, 4.565651165351640, 4.879617732141908, 5.351137689295675, 6.117761815137711, 6.572012297769611, 7.159357425304693, 7.675499356935150, 8.033768032792740, 8.430753378082330, 8.810870422495194, 9.353694097974730, 9.973280916590532, 10.34878345091586, 10.73677793303801, 11.13836686147809, 11.79990814961824, 12.14702105344254, 12.55727179087527
