| L(s) = 1 | + 3-s − 4-s − 7-s − 12-s − 2·13-s + 19-s − 21-s − 27-s + 28-s + 4·31-s + 2·37-s − 2·39-s − 43-s + 49-s + 2·52-s + 57-s − 2·61-s + 64-s − 67-s + 2·73-s − 76-s − 2·79-s − 81-s + 84-s + 2·91-s + 4·93-s + 2·97-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s − 7-s − 12-s − 2·13-s + 19-s − 21-s − 27-s + 28-s + 4·31-s + 2·37-s − 2·39-s − 43-s + 49-s + 2·52-s + 57-s − 2·61-s + 64-s − 67-s + 2·73-s − 76-s − 2·79-s − 81-s + 84-s + 2·91-s + 4·93-s + 2·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7795786764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7795786764\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−10.14481120059794087714542727938, −9.852246864281772762571094234387, −9.610824892434926778972806506377, −9.184768914145084891096575728697, −8.926594045490684485566430927972, −8.223345515667233378018295425003, −8.143038880934048142478579951979, −7.41785223496469855228093816686, −7.37145874042251345594491716672, −6.54631896126174916189396117275, −6.20060069090052131309275776188, −5.73813529112427156566665246733, −4.92930580491230472922330850948, −4.57442467153360372299369917498, −4.46368752189094048442954809538, −3.51420551010126304392432655651, −3.04054851144198347176414363508, −2.70408263627851848195748200723, −2.19207251486768035259803088450, −0.813275479504411650719544841679,
0.813275479504411650719544841679, 2.19207251486768035259803088450, 2.70408263627851848195748200723, 3.04054851144198347176414363508, 3.51420551010126304392432655651, 4.46368752189094048442954809538, 4.57442467153360372299369917498, 4.92930580491230472922330850948, 5.73813529112427156566665246733, 6.20060069090052131309275776188, 6.54631896126174916189396117275, 7.37145874042251345594491716672, 7.41785223496469855228093816686, 8.143038880934048142478579951979, 8.223345515667233378018295425003, 8.926594045490684485566430927972, 9.184768914145084891096575728697, 9.610824892434926778972806506377, 9.852246864281772762571094234387, 10.14481120059794087714542727938
