| L(s) = 1 | + 2·2-s + 4·4-s + 3·5-s − 7·7-s + 8·8-s + 6·10-s + 23·11-s − 13·13-s − 14·14-s + 16·16-s − 133·17-s + 103·19-s + 12·20-s + 46·22-s − 113·23-s − 116·25-s − 26·26-s − 28·28-s + 141·29-s − 90·31-s + 32·32-s − 266·34-s − 21·35-s − 233·37-s + 206·38-s + 24·40-s + 230·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.268·5-s − 0.377·7-s + 0.353·8-s + 0.189·10-s + 0.630·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 1.89·17-s + 1.24·19-s + 0.134·20-s + 0.445·22-s − 1.02·23-s − 0.927·25-s − 0.196·26-s − 0.188·28-s + 0.902·29-s − 0.521·31-s + 0.176·32-s − 1.34·34-s − 0.101·35-s − 1.03·37-s + 0.879·38-s + 0.0948·40-s + 0.876·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 23 T + p^{3} T^{2} \) |
| 17 | \( 1 + 133 T + p^{3} T^{2} \) |
| 19 | \( 1 - 103 T + p^{3} T^{2} \) |
| 23 | \( 1 + 113 T + p^{3} T^{2} \) |
| 29 | \( 1 - 141 T + p^{3} T^{2} \) |
| 31 | \( 1 + 90 T + p^{3} T^{2} \) |
| 37 | \( 1 + 233 T + p^{3} T^{2} \) |
| 41 | \( 1 - 230 T + p^{3} T^{2} \) |
| 43 | \( 1 + 337 T + p^{3} T^{2} \) |
| 47 | \( 1 + 96 T + p^{3} T^{2} \) |
| 53 | \( 1 - 134 T + p^{3} T^{2} \) |
| 59 | \( 1 + 838 T + p^{3} T^{2} \) |
| 61 | \( 1 - 295 T + p^{3} T^{2} \) |
| 67 | \( 1 - 468 T + p^{3} T^{2} \) |
| 71 | \( 1 + 54 T + p^{3} T^{2} \) |
| 73 | \( 1 + 553 T + p^{3} T^{2} \) |
| 79 | \( 1 + 694 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1010 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 102 T + p^{3} T^{2} \) |
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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
−8.683873277688106208236692179069, −7.65312471797779454313371998128, −6.78158959526073890187082588950, −6.22635517474770263868187982453, −5.30633838622752582594638388038, −4.39746053764922901431023705925, −3.60429408542670009897385182137, −2.53372265493715277455485768419, −1.58610746263770701786178326573, 0,
1.58610746263770701786178326573, 2.53372265493715277455485768419, 3.60429408542670009897385182137, 4.39746053764922901431023705925, 5.30633838622752582594638388038, 6.22635517474770263868187982453, 6.78158959526073890187082588950, 7.65312471797779454313371998128, 8.683873277688106208236692179069
