| L(s) = 1 | − 4-s + 2·7-s + 16-s − 2·25-s − 2·28-s + 3·49-s − 64-s − 4·79-s + 2·100-s + 2·112-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 4·175-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4-s + 2·7-s + 16-s − 2·25-s − 2·28-s + 3·49-s − 64-s − 4·79-s + 2·100-s + 2·112-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 4·175-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7387514190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7387514190\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$ | \( ( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−11.47242509949491631579469741844, −11.02736288555510689890473445734, −10.35107207482332204488348574358, −10.07024174942785458389292631485, −9.627977813884985640099135376941, −9.027915968570364710771121702588, −8.534015845060712995438600939245, −8.402449026000533424865704733239, −7.67717943805798144423115023471, −7.61542465395263608247968716370, −6.97513040651102840211018154660, −5.95012340013087900626517670500, −5.79300872100055039029290891748, −5.13862652968003523422956471952, −4.75866191754171657278325285436, −4.11112593998298205424011014724, −3.92634248655435326341959254362, −2.89421827024905918171286179056, −1.98369331142917368553873576877, −1.34936159779161659946630610031,
1.34936159779161659946630610031, 1.98369331142917368553873576877, 2.89421827024905918171286179056, 3.92634248655435326341959254362, 4.11112593998298205424011014724, 4.75866191754171657278325285436, 5.13862652968003523422956471952, 5.79300872100055039029290891748, 5.95012340013087900626517670500, 6.97513040651102840211018154660, 7.61542465395263608247968716370, 7.67717943805798144423115023471, 8.402449026000533424865704733239, 8.534015845060712995438600939245, 9.027915968570364710771121702588, 9.627977813884985640099135376941, 10.07024174942785458389292631485, 10.35107207482332204488348574358, 11.02736288555510689890473445734, 11.47242509949491631579469741844
