| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s − 2·9-s + 2·12-s − 4·16-s + 4·18-s − 5·27-s + 16·31-s + 8·32-s − 4·36-s − 4·48-s − 10·49-s − 8·53-s + 10·54-s − 32·62-s − 8·64-s + 20·79-s + 81-s + 18·83-s + 16·93-s + 8·96-s + 20·98-s + 16·106-s − 34·107-s − 10·108-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s − 2/3·9-s + 0.577·12-s − 16-s + 0.942·18-s − 0.962·27-s + 2.87·31-s + 1.41·32-s − 2/3·36-s − 0.577·48-s − 1.42·49-s − 1.09·53-s + 1.36·54-s − 4.06·62-s − 64-s + 2.25·79-s + 1/9·81-s + 1.97·83-s + 1.65·93-s + 0.816·96-s + 2.02·98-s + 1.55·106-s − 3.28·107-s − 0.962·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8592417278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8592417278\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.665421196412280182170334766998, −8.182910838086186172188996077240, −7.974950820166071766849701202962, −7.77921455567468453333623630561, −6.88699879047458142537014880578, −6.54282764581716256623518288017, −6.19421351777184295505297292819, −5.39258018193498154600079656616, −4.80562532332567230387554534751, −4.35480286006529544992633071822, −3.52151660613402712389144369009, −2.90862550946813728509296480615, −2.38035753805067884592817569239, −1.61243686764748739595327097028, −0.64659822666269902998848780560,
0.64659822666269902998848780560, 1.61243686764748739595327097028, 2.38035753805067884592817569239, 2.90862550946813728509296480615, 3.52151660613402712389144369009, 4.35480286006529544992633071822, 4.80562532332567230387554534751, 5.39258018193498154600079656616, 6.19421351777184295505297292819, 6.54282764581716256623518288017, 6.88699879047458142537014880578, 7.77921455567468453333623630561, 7.974950820166071766849701202962, 8.182910838086186172188996077240, 8.665421196412280182170334766998
