| L(s) = 1 | + 5-s − 9-s + 2·11-s − 2·31-s − 45-s − 2·49-s + 2·55-s + 2·59-s + 2·71-s + 2·89-s − 2·99-s + 3·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·155-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 5-s − 9-s + 2·11-s − 2·31-s − 45-s − 2·49-s + 2·55-s + 2·59-s + 2·71-s + 2·89-s − 2·99-s + 3·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·155-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 774400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 774400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.142934288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.142934288\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−10.64885043816639158555759927902, −9.816064417801211431424999679181, −9.777571143693556337217260708940, −9.241654603129487465488687566121, −9.075804793792927686631635357774, −8.448033777754738876352687834600, −8.275323252312199843062904256573, −7.50702262269627597452826470860, −7.05460101415433733235848336056, −6.61612082254370158783954504007, −6.15893898753683633052192165913, −5.93806298026957659300460429578, −5.29324465123950895712765949971, −4.99592629166217049725002194104, −4.21205817982189336098735354918, −3.52942851288700712550693422582, −3.47037170613842761347265608124, −2.38218283205467525109772655125, −1.96335331163337255569281907212, −1.22620975675728170239341745463,
1.22620975675728170239341745463, 1.96335331163337255569281907212, 2.38218283205467525109772655125, 3.47037170613842761347265608124, 3.52942851288700712550693422582, 4.21205817982189336098735354918, 4.99592629166217049725002194104, 5.29324465123950895712765949971, 5.93806298026957659300460429578, 6.15893898753683633052192165913, 6.61612082254370158783954504007, 7.05460101415433733235848336056, 7.50702262269627597452826470860, 8.275323252312199843062904256573, 8.448033777754738876352687834600, 9.075804793792927686631635357774, 9.241654603129487465488687566121, 9.777571143693556337217260708940, 9.816064417801211431424999679181, 10.64885043816639158555759927902
