| L(s) = 1 | + 118·11-s − 218·19-s − 64·29-s + 20·31-s − 234·41-s + 10·49-s + 784·59-s − 1.42e3·61-s + 1.22e3·71-s − 828·79-s − 162·89-s + 468·101-s + 2.46e3·109-s + 7.78e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.61e3·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 3.23·11-s − 2.63·19-s − 0.409·29-s + 0.115·31-s − 0.891·41-s + 0.0291·49-s + 1.72·59-s − 2.98·61-s + 2.04·71-s − 1.17·79-s − 0.192·89-s + 0.461·101-s + 2.16·109-s + 5.84·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.64·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.982906474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.982906474\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 59 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3610 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9801 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 109 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 13302 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 62102 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 117 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 8470 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 203022 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 297430 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 392 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 710 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 537517 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 612 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 476633 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 414 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1128933 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 81 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 430658 T^{2} + p^{6} T^{4} \) |
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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
−9.169232081849544560258080886701, −8.739352462117783253898482797103, −8.323365778971144537002632736119, −8.247445742291236507989237138225, −7.35167160520441183579162092696, −7.02033800461129040265028467453, −6.71146878187849136172864981051, −6.31348728095649617895750994294, −6.08674094104886790281605855180, −5.69881370139695008382192993839, −4.71916280837525754539555504184, −4.64232577711783517701036301902, −4.04216965202331267021392616582, −3.78142992236871776282432252020, −3.41715714102425631889962986906, −2.62586434301836972250073444277, −1.82564185894653098058708160546, −1.78540450460263149686473975294, −1.04316623211780955147944849462, −0.39580808936499960434353635943,
0.39580808936499960434353635943, 1.04316623211780955147944849462, 1.78540450460263149686473975294, 1.82564185894653098058708160546, 2.62586434301836972250073444277, 3.41715714102425631889962986906, 3.78142992236871776282432252020, 4.04216965202331267021392616582, 4.64232577711783517701036301902, 4.71916280837525754539555504184, 5.69881370139695008382192993839, 6.08674094104886790281605855180, 6.31348728095649617895750994294, 6.71146878187849136172864981051, 7.02033800461129040265028467453, 7.35167160520441183579162092696, 8.247445742291236507989237138225, 8.323365778971144537002632736119, 8.739352462117783253898482797103, 9.169232081849544560258080886701
