| L(s) = 1 | + 2-s + 4-s + 8-s + 6·11-s + 4·13-s + 16-s + 6·22-s + 3·23-s − 4·25-s + 4·26-s + 32-s + 7·37-s + 6·44-s + 3·46-s − 3·47-s − 4·49-s − 4·50-s + 4·52-s − 6·59-s − 17·61-s + 64-s + 9·71-s − 2·73-s + 7·74-s − 3·83-s + 6·88-s + 3·92-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s + 1.10·13-s + 1/4·16-s + 1.27·22-s + 0.625·23-s − 4/5·25-s + 0.784·26-s + 0.176·32-s + 1.15·37-s + 0.904·44-s + 0.442·46-s − 0.437·47-s − 4/7·49-s − 0.565·50-s + 0.554·52-s − 0.781·59-s − 2.17·61-s + 1/8·64-s + 1.06·71-s − 0.234·73-s + 0.813·74-s − 0.329·83-s + 0.639·88-s + 0.312·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.253113343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.253113343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.030996989438781728130189917057, −8.736475026258367752049914121628, −7.925307182018892621669287036291, −7.72249246460450028114927387731, −6.93006545507709582849218480790, −6.54665552175735329547565616894, −6.09892175179055918410560684533, −5.83132152023505480827070717523, −4.95690164496798987799026333370, −4.47279346799588355319748628621, −3.89877096820186001865602660196, −3.50118123062177251535335909911, −2.82677571691491076274597392456, −1.78869371510594636858718678253, −1.17272335587756497254229298370,
1.17272335587756497254229298370, 1.78869371510594636858718678253, 2.82677571691491076274597392456, 3.50118123062177251535335909911, 3.89877096820186001865602660196, 4.47279346799588355319748628621, 4.95690164496798987799026333370, 5.83132152023505480827070717523, 6.09892175179055918410560684533, 6.54665552175735329547565616894, 6.93006545507709582849218480790, 7.72249246460450028114927387731, 7.925307182018892621669287036291, 8.736475026258367752049914121628, 9.030996989438781728130189917057
