| L(s) = 1 | − 2·5-s − 6·9-s + 8·19-s + 5·25-s − 8·29-s + 16·31-s + 24·41-s + 12·45-s − 8·59-s + 12·61-s + 48·71-s − 8·79-s + 9·81-s − 20·89-s − 16·95-s + 36·101-s + 28·109-s + 22·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s − 32·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 2·9-s + 1.83·19-s + 25-s − 1.48·29-s + 2.87·31-s + 3.74·41-s + 1.78·45-s − 1.04·59-s + 1.53·61-s + 5.69·71-s − 0.900·79-s + 81-s − 2.11·89-s − 1.64·95-s + 3.58·101-s + 2.68·109-s + 2·121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s − 2.57·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.634451288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.634451288\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 18 T^{2} - 205 T^{4} - 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.27448955108484871978425618953, −6.77258980269466721507198350982, −6.76584509139964317939262838027, −6.63730428256926712623220216119, −6.14930804088332026789838022708, −5.89815279966537972382874222445, −5.84734254600871668037574197444, −5.70928072489352256982493990613, −5.29699790414913958053695680384, −5.07194342243141494535938170370, −4.96998853271478763571430683095, −4.53978395344829857762746738944, −4.41524035427151186101550105026, −4.06095558429288036321659118680, −3.77414100600725362662326140358, −3.47530894235330696219250094753, −3.37635752384361742849959081639, −2.78372263240156959442536848385, −2.78359223102205622521061339146, −2.70595075234222469454391472969, −2.05542799489296728166079430194, −1.93231285190119175792442378616, −0.962554427255944045818718986399, −0.845848652453346445710810900749, −0.55861095119285999194782798143,
0.55861095119285999194782798143, 0.845848652453346445710810900749, 0.962554427255944045818718986399, 1.93231285190119175792442378616, 2.05542799489296728166079430194, 2.70595075234222469454391472969, 2.78359223102205622521061339146, 2.78372263240156959442536848385, 3.37635752384361742849959081639, 3.47530894235330696219250094753, 3.77414100600725362662326140358, 4.06095558429288036321659118680, 4.41524035427151186101550105026, 4.53978395344829857762746738944, 4.96998853271478763571430683095, 5.07194342243141494535938170370, 5.29699790414913958053695680384, 5.70928072489352256982493990613, 5.84734254600871668037574197444, 5.89815279966537972382874222445, 6.14930804088332026789838022708, 6.63730428256926712623220216119, 6.76584509139964317939262838027, 6.77258980269466721507198350982, 7.27448955108484871978425618953
