| L(s) = 1 | + 2-s + 2·3-s − 4-s − 5-s + 2·6-s + 2·7-s − 3·8-s + 9-s − 10-s − 2·12-s + 13-s + 2·14-s − 2·15-s − 16-s − 5·17-s + 18-s + 6·19-s + 20-s + 4·21-s − 2·23-s − 6·24-s − 4·25-s + 26-s − 4·27-s − 2·28-s + 9·29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.577·12-s + 0.277·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s − 1.21·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.872·21-s − 0.417·23-s − 1.22·24-s − 4/5·25-s + 0.196·26-s − 0.769·27-s − 0.377·28-s + 1.67·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.191331063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.191331063\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 37 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.40349796016596, −13.00491279409493, −12.31800476513075, −11.85749650959503, −11.54552878719847, −11.03156756423408, −10.33297704443279, −9.702062488228433, −9.399035852837460, −8.772081818265702, −8.419713110463409, −8.101905611842933, −7.566962584942820, −7.012322144351054, −6.271204071675044, −5.817232967657762, −5.148281758719087, −4.666893542525455, −4.206398570208976, −3.711049365338597, −3.233380326399667, −2.592993588355693, −2.178926113733892, −1.255056717363337, −0.5085687946487075,
0.5085687946487075, 1.255056717363337, 2.178926113733892, 2.592993588355693, 3.233380326399667, 3.711049365338597, 4.206398570208976, 4.666893542525455, 5.148281758719087, 5.817232967657762, 6.271204071675044, 7.012322144351054, 7.566962584942820, 8.101905611842933, 8.419713110463409, 8.772081818265702, 9.399035852837460, 9.702062488228433, 10.33297704443279, 11.03156756423408, 11.54552878719847, 11.85749650959503, 12.31800476513075, 13.00491279409493, 13.40349796016596
