| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 5·11-s − 6·13-s + 14-s + 16-s + 17-s − 3·19-s − 5·22-s + 6·26-s − 28-s + 6·29-s − 4·31-s − 32-s − 34-s + 8·37-s + 3·38-s − 11·41-s − 8·43-s + 5·44-s − 2·47-s + 49-s − 6·52-s − 4·53-s + 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.50·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.688·19-s − 1.06·22-s + 1.17·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.171·34-s + 1.31·37-s + 0.486·38-s − 1.71·41-s − 1.21·43-s + 0.753·44-s − 0.291·47-s + 1/7·49-s − 0.832·52-s − 0.549·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 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
−8.387182329094963452351149311648, −7.60593169459124745854077628187, −6.71238157904374327273470767010, −6.46479582570974087018240392764, −5.26015307963936998972992039268, −4.39148677091846193085504832465, −3.40111091922544740626092357501, −2.42817721651914966878410253933, −1.39287136921494160793057743506, 0,
1.39287136921494160793057743506, 2.42817721651914966878410253933, 3.40111091922544740626092357501, 4.39148677091846193085504832465, 5.26015307963936998972992039268, 6.46479582570974087018240392764, 6.71238157904374327273470767010, 7.60593169459124745854077628187, 8.387182329094963452351149311648
