| L(s) = 1 | + 2·5-s + 2·9-s − 8·11-s + 8·19-s − 25-s + 4·29-s + 4·41-s + 4·45-s + 10·49-s − 16·55-s + 24·59-s + 20·61-s − 16·71-s + 32·79-s − 5·81-s − 12·89-s + 16·95-s − 16·99-s − 12·101-s − 12·109-s + 26·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2/3·9-s − 2.41·11-s + 1.83·19-s − 1/5·25-s + 0.742·29-s + 0.624·41-s + 0.596·45-s + 10/7·49-s − 2.15·55-s + 3.12·59-s + 2.56·61-s − 1.89·71-s + 3.60·79-s − 5/9·81-s − 1.27·89-s + 1.64·95-s − 1.60·99-s − 1.19·101-s − 1.14·109-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.722474975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.722474975\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−11.70788594371610619857230734194, −11.56003040110246279599858175777, −10.67417227178092479248836645418, −10.35601431790166315181169347920, −10.18685347995366398591103840290, −9.442278828225751355671888691640, −9.439291448262045443523774789712, −8.286652205325374798821505754513, −8.244666969700496587120338990414, −7.49390492065617882517364478080, −7.16092932639731528465211978098, −6.61140314742272237450934732165, −5.67291059682368933672061423358, −5.41765569838809309218604951719, −5.15679754884998928140906335902, −4.27704939201964139077046919386, −3.51247017092291311420608862381, −2.57721743382362538607804729472, −2.33249497133181333909952594374, −1.00408633879289146220868726536,
1.00408633879289146220868726536, 2.33249497133181333909952594374, 2.57721743382362538607804729472, 3.51247017092291311420608862381, 4.27704939201964139077046919386, 5.15679754884998928140906335902, 5.41765569838809309218604951719, 5.67291059682368933672061423358, 6.61140314742272237450934732165, 7.16092932639731528465211978098, 7.49390492065617882517364478080, 8.244666969700496587120338990414, 8.286652205325374798821505754513, 9.439291448262045443523774789712, 9.442278828225751355671888691640, 10.18685347995366398591103840290, 10.35601431790166315181169347920, 10.67417227178092479248836645418, 11.56003040110246279599858175777, 11.70788594371610619857230734194
