| L(s) = 1 | + 4·5-s + 2·9-s + 16·19-s + 11·25-s + 10·29-s + 10·31-s − 14·41-s + 8·45-s + 13·49-s + 6·59-s − 12·61-s − 26·71-s − 28·79-s − 5·81-s + 28·89-s + 64·95-s + 30·101-s − 36·109-s − 22·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 40·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2/3·9-s + 3.67·19-s + 11/5·25-s + 1.85·29-s + 1.79·31-s − 2.18·41-s + 1.19·45-s + 13/7·49-s + 0.781·59-s − 1.53·61-s − 3.08·71-s − 3.15·79-s − 5/9·81-s + 2.96·89-s + 6.56·95-s + 2.98·101-s − 3.44·109-s − 2·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.163173229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.163173229\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.576148030410108098600644602424, −8.951532487729763002332319976483, −8.937412046044829089611044227641, −8.375765579340527809822076059588, −7.76313551498356507799038903369, −7.43975023962131211940246693187, −7.00935219860697231240192946526, −6.74325802401862703672458195401, −6.09269721317024906131007890852, −5.95137489070646669466531836892, −5.33328016647002968014205843757, −5.09448648118850537508863782361, −4.70235207395148745835275342817, −4.18722899653004125931631119656, −3.32978838267781671944453397696, −2.84274349368173337774974213154, −2.81783495171724879901785858667, −1.83492881296117993969960863091, −1.17306808228243964742902952309, −1.07977648410735711665803284198,
1.07977648410735711665803284198, 1.17306808228243964742902952309, 1.83492881296117993969960863091, 2.81783495171724879901785858667, 2.84274349368173337774974213154, 3.32978838267781671944453397696, 4.18722899653004125931631119656, 4.70235207395148745835275342817, 5.09448648118850537508863782361, 5.33328016647002968014205843757, 5.95137489070646669466531836892, 6.09269721317024906131007890852, 6.74325802401862703672458195401, 7.00935219860697231240192946526, 7.43975023962131211940246693187, 7.76313551498356507799038903369, 8.375765579340527809822076059588, 8.937412046044829089611044227641, 8.951532487729763002332319976483, 9.576148030410108098600644602424
