| L(s) = 1 | + 8·7-s − 9-s − 4·17-s − 16·23-s − 25-s − 20·41-s − 16·47-s + 34·49-s − 8·63-s + 28·73-s − 32·79-s + 81-s − 4·89-s + 4·97-s + 8·103-s + 12·113-s − 32·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + 157-s − 128·161-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 1/3·9-s − 0.970·17-s − 3.33·23-s − 1/5·25-s − 3.12·41-s − 2.33·47-s + 34/7·49-s − 1.00·63-s + 3.27·73-s − 3.60·79-s + 1/9·81-s − 0.423·89-s + 0.406·97-s + 0.788·103-s + 1.12·113-s − 2.93·119-s + 2·121-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 + 0.323·153-s + 0.0798·157-s − 10.0·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.955322832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.955322832\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.631708038465271509636149152327, −8.300104031527466841121200382345, −8.254725105928943268422837281103, −7.60063200755130718247577608228, −7.42998395045456739396150145840, −6.82654893591999681737315619370, −6.49170980420905646347835011813, −5.97813839942630910510714627822, −5.64187712399003463256000855771, −5.24683298351127885171103339580, −4.80718631124847877747972853011, −4.52925597230107979277564716679, −4.35416871791474225160972864503, −3.60331562053031336699909715989, −3.43008231842520856884765082610, −2.44424211886591476756172329866, −2.04407006503562681316775408195, −1.68427033679691729673500219140, −1.57224830848052748165244866993, −0.37767978696971425867334090635,
0.37767978696971425867334090635, 1.57224830848052748165244866993, 1.68427033679691729673500219140, 2.04407006503562681316775408195, 2.44424211886591476756172329866, 3.43008231842520856884765082610, 3.60331562053031336699909715989, 4.35416871791474225160972864503, 4.52925597230107979277564716679, 4.80718631124847877747972853011, 5.24683298351127885171103339580, 5.64187712399003463256000855771, 5.97813839942630910510714627822, 6.49170980420905646347835011813, 6.82654893591999681737315619370, 7.42998395045456739396150145840, 7.60063200755130718247577608228, 8.254725105928943268422837281103, 8.300104031527466841121200382345, 8.631708038465271509636149152327
