| L(s) = 1 | + 4·5-s + 2·9-s − 8·19-s + 11·25-s + 18·29-s + 6·31-s + 18·41-s + 8·45-s + 5·49-s + 18·59-s − 4·61-s + 26·71-s − 4·79-s − 5·81-s + 20·89-s − 32·95-s − 26·101-s + 4·109-s − 22·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 72·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 2/3·9-s − 1.83·19-s + 11/5·25-s + 3.34·29-s + 1.07·31-s + 2.81·41-s + 1.19·45-s + 5/7·49-s + 2.34·59-s − 0.512·61-s + 3.08·71-s − 0.450·79-s − 5/9·81-s + 2.11·89-s − 3.28·95-s − 2.58·101-s + 0.383·109-s − 2·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.97·145-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
| \(L(1)\) |
\(\approx\) |
\(4.551777782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.551777782\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.891853816327565069451934964429, −9.106880328259528099367033204138, −8.583204717326241163457665832685, −8.454109571540342954702242546988, −8.014136237444527536420658788535, −7.41921580403427661255451879341, −6.86338147016201561933453114650, −6.55897312714781837698762306884, −6.20508359996827198648627932013, −6.14661255766429408260920904217, −5.27769580530086317560682466396, −5.13324573187851431949735853493, −4.36357284262363655520652507468, −4.36214722805439604445009313855, −3.63720928975287075157645829245, −2.69073401660752880830005209203, −2.44472970356902299281897618415, −2.24498697136300666590928618800, −1.16587171903868198005473538032, −0.947255169095409542204540926973,
0.947255169095409542204540926973, 1.16587171903868198005473538032, 2.24498697136300666590928618800, 2.44472970356902299281897618415, 2.69073401660752880830005209203, 3.63720928975287075157645829245, 4.36214722805439604445009313855, 4.36357284262363655520652507468, 5.13324573187851431949735853493, 5.27769580530086317560682466396, 6.14661255766429408260920904217, 6.20508359996827198648627932013, 6.55897312714781837698762306884, 6.86338147016201561933453114650, 7.41921580403427661255451879341, 8.014136237444527536420658788535, 8.454109571540342954702242546988, 8.583204717326241163457665832685, 9.106880328259528099367033204138, 9.891853816327565069451934964429
