| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s − 14·7-s + 8·8-s + 10·10-s + 36·11-s − 13·13-s − 28·14-s + 16·16-s − 68·17-s − 158·19-s + 20·20-s + 72·22-s − 46·23-s + 25·25-s − 26·26-s − 56·28-s + 8·29-s − 176·31-s + 32·32-s − 136·34-s − 70·35-s + 62·37-s − 316·38-s + 40·40-s − 30·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s + 0.986·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.970·17-s − 1.90·19-s + 0.223·20-s + 0.697·22-s − 0.417·23-s + 1/5·25-s − 0.196·26-s − 0.377·28-s + 0.0512·29-s − 1.01·31-s + 0.176·32-s − 0.685·34-s − 0.338·35-s + 0.275·37-s − 1.34·38-s + 0.158·40-s − 0.114·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 7 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 17 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 158 T + p^{3} T^{2} \) |
| 23 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 8 T + p^{3} T^{2} \) |
| 31 | \( 1 + 176 T + p^{3} T^{2} \) |
| 37 | \( 1 - 62 T + p^{3} T^{2} \) |
| 41 | \( 1 + 30 T + p^{3} T^{2} \) |
| 43 | \( 1 - 252 T + p^{3} T^{2} \) |
| 47 | \( 1 - 120 T + p^{3} T^{2} \) |
| 53 | \( 1 + 758 T + p^{3} T^{2} \) |
| 59 | \( 1 + 252 T + p^{3} T^{2} \) |
| 61 | \( 1 - 398 T + p^{3} T^{2} \) |
| 67 | \( 1 - 884 T + p^{3} T^{2} \) |
| 71 | \( 1 - 80 T + p^{3} T^{2} \) |
| 73 | \( 1 + 660 T + p^{3} T^{2} \) |
| 79 | \( 1 - 568 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1084 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1250 T + p^{3} T^{2} \) |
| 97 | \( 1 - 84 T + p^{3} T^{2} \) |
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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
−9.120654269946622715533171111120, −8.201574173970765119757494641500, −6.89132663759917562542818537918, −6.48554916623995144888590839669, −5.71355042188842820665910704881, −4.49335957464199641763659431171, −3.85303919219020445718625563758, −2.64134950042028064734802430644, −1.71673117453714557135331965772, 0,
1.71673117453714557135331965772, 2.64134950042028064734802430644, 3.85303919219020445718625563758, 4.49335957464199641763659431171, 5.71355042188842820665910704881, 6.48554916623995144888590839669, 6.89132663759917562542818537918, 8.201574173970765119757494641500, 9.120654269946622715533171111120
