| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 3·9-s + 4·11-s − 2·13-s − 2·14-s + 16-s − 17-s + 3·18-s − 4·19-s − 4·22-s − 6·23-s − 5·25-s + 2·26-s + 2·28-s − 6·31-s − 32-s + 34-s − 3·36-s + 8·37-s + 4·38-s + 6·41-s + 4·43-s + 4·44-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 9-s + 1.20·11-s − 0.554·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.707·18-s − 0.917·19-s − 0.852·22-s − 1.25·23-s − 25-s + 0.392·26-s + 0.377·28-s − 1.07·31-s − 0.176·32-s + 0.171·34-s − 1/2·36-s + 1.31·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s + 0.603·44-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.790693952044951997478328380268, −8.049113476362195302642519575432, −7.45891562208992574712782831414, −6.31226176693689050619220239120, −5.87118436970780233730782712461, −4.63523940958072315693785643370, −3.76887213530047460241739819058, −2.46597311586538152207516771903, −1.61195577324018909918671478287, 0,
1.61195577324018909918671478287, 2.46597311586538152207516771903, 3.76887213530047460241739819058, 4.63523940958072315693785643370, 5.87118436970780233730782712461, 6.31226176693689050619220239120, 7.45891562208992574712782831414, 8.049113476362195302642519575432, 8.790693952044951997478328380268
