| L(s) = 1 | − 8·7-s − 8·9-s + 8·25-s − 16·31-s − 48·47-s + 34·49-s + 64·63-s + 30·81-s + 32·103-s + 48·113-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s − 64·175-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 8/3·9-s + 8/5·25-s − 2.87·31-s − 7.00·47-s + 34/7·49-s + 8.06·63-s + 10/3·81-s + 3.15·103-s + 4.51·113-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 0.0760·173-s − 4.83·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3624031154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3624031154\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 164 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.47633292969634126639120695876, −6.30543652740804667966090025488, −6.26908018765110978031386408301, −6.09915389211888899443570586443, −5.81219448872631636487646428997, −5.58722683983827035671839617782, −5.43746306319858820469356948071, −4.98560089238178609163330829418, −4.97977007659164436292842963849, −4.89500337509889590491736832549, −4.48880444144912288992086597411, −4.02167549283901871425755623771, −3.73385122164286371453086305277, −3.55659164227034875494149926376, −3.37604643313068090805398346571, −3.18462022847113848654945723389, −2.94305814379119032091178416723, −2.87619698558207936072158024141, −2.80960545803790820757749456869, −1.98958555608426959772381761468, −1.88454384638943013246555856575, −1.80223123193085840045483628597, −0.944766728991046706432242748997, −0.37453727038304773158592530875, −0.25733223638398510423697810450,
0.25733223638398510423697810450, 0.37453727038304773158592530875, 0.944766728991046706432242748997, 1.80223123193085840045483628597, 1.88454384638943013246555856575, 1.98958555608426959772381761468, 2.80960545803790820757749456869, 2.87619698558207936072158024141, 2.94305814379119032091178416723, 3.18462022847113848654945723389, 3.37604643313068090805398346571, 3.55659164227034875494149926376, 3.73385122164286371453086305277, 4.02167549283901871425755623771, 4.48880444144912288992086597411, 4.89500337509889590491736832549, 4.97977007659164436292842963849, 4.98560089238178609163330829418, 5.43746306319858820469356948071, 5.58722683983827035671839617782, 5.81219448872631636487646428997, 6.09915389211888899443570586443, 6.26908018765110978031386408301, 6.30543652740804667966090025488, 6.47633292969634126639120695876
