| L(s) = 1 | + 4·7-s + 4·25-s − 32·37-s + 8·43-s − 2·49-s − 32·67-s + 16·79-s + 16·109-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4·169-s + 173-s + 16·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 4/5·25-s − 5.26·37-s + 1.21·43-s − 2/7·49-s − 3.90·67-s + 1.80·79-s + 1.53·109-s + 8/11·121-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 + 4/13·169-s + 0.0760·173-s + 1.20·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8180684839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8180684839\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 124 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.27586051529130055199366605375, −7.10842077039591159762704676526, −6.68103600269940484301036404218, −6.40570421645634995482108107871, −6.23999886810543594180989090665, −6.10305942880796060122452535495, −5.71043500669469085099172913757, −5.47631116130051516878113645504, −5.26187614450275704637581972518, −5.01274627049784423938302269229, −4.76376120542697806171471507490, −4.73190106636080514521640132173, −4.54290578387824164368309994845, −3.97907493330040818301301053535, −3.87810241512098645269686149545, −3.32828193221980080036557981408, −3.30281762692379705279491977101, −3.30250079332764097116761737800, −2.52656087614917534912737501383, −2.23099435967430749903964436616, −2.19819323526895105598158560666, −1.51454079073503463902057065716, −1.41552142900784671795552119596, −1.20595817793977360737787846470, −0.19296566215816578712274547067,
0.19296566215816578712274547067, 1.20595817793977360737787846470, 1.41552142900784671795552119596, 1.51454079073503463902057065716, 2.19819323526895105598158560666, 2.23099435967430749903964436616, 2.52656087614917534912737501383, 3.30250079332764097116761737800, 3.30281762692379705279491977101, 3.32828193221980080036557981408, 3.87810241512098645269686149545, 3.97907493330040818301301053535, 4.54290578387824164368309994845, 4.73190106636080514521640132173, 4.76376120542697806171471507490, 5.01274627049784423938302269229, 5.26187614450275704637581972518, 5.47631116130051516878113645504, 5.71043500669469085099172913757, 6.10305942880796060122452535495, 6.23999886810543594180989090665, 6.40570421645634995482108107871, 6.68103600269940484301036404218, 7.10842077039591159762704676526, 7.27586051529130055199366605375
