| L(s) = 1 | + 4·5-s − 9-s − 4·11-s − 8·19-s + 11·25-s − 8·29-s − 16·31-s − 12·41-s − 4·45-s + 10·49-s − 16·55-s + 20·59-s − 28·61-s + 8·71-s − 16·79-s + 81-s − 12·89-s − 32·95-s + 4·99-s + 8·101-s + 36·109-s − 10·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s − 1.20·11-s − 1.83·19-s + 11/5·25-s − 1.48·29-s − 2.87·31-s − 1.87·41-s − 0.596·45-s + 10/7·49-s − 2.15·55-s + 2.60·59-s − 3.58·61-s + 0.949·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s − 3.28·95-s + 0.402·99-s + 0.796·101-s + 3.44·109-s − 0.909·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.040961068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040961068\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.133268811439705425744653861659, −8.603798841262593642195515003299, −8.344736712617183097635433907133, −7.57750170309230984537305192346, −7.45040171591637541988342342090, −6.89687004795153531334743827776, −6.71509816029360762341239124059, −5.94947025571532300547259944484, −5.86624795008074368619694933757, −5.55896621300738060861822604438, −5.21686885616493223095949956927, −4.72234377214669647550459857644, −4.31890707998252421772462088689, −3.51500931558475101007930422432, −3.43088109584937124432054037674, −2.53329523386891776614274580468, −2.32225797222956469181091582995, −1.83546881909555737139101018036, −1.52987831900190524830934529347, −0.28491494573742795891265509846,
0.28491494573742795891265509846, 1.52987831900190524830934529347, 1.83546881909555737139101018036, 2.32225797222956469181091582995, 2.53329523386891776614274580468, 3.43088109584937124432054037674, 3.51500931558475101007930422432, 4.31890707998252421772462088689, 4.72234377214669647550459857644, 5.21686885616493223095949956927, 5.55896621300738060861822604438, 5.86624795008074368619694933757, 5.94947025571532300547259944484, 6.71509816029360762341239124059, 6.89687004795153531334743827776, 7.45040171591637541988342342090, 7.57750170309230984537305192346, 8.344736712617183097635433907133, 8.603798841262593642195515003299, 9.133268811439705425744653861659
