| L(s) = 1 | + 2·2-s − 8·5-s − 8·8-s − 16·10-s − 44·13-s − 16·16-s + 22·17-s − 2·25-s − 88·26-s − 68·29-s + 44·34-s − 32·37-s + 64·40-s − 26·41-s + 86·49-s − 4·50-s − 104·53-s − 136·58-s − 32·61-s + 64·64-s + 352·65-s − 50·73-s − 64·74-s + 128·80-s − 52·82-s − 176·85-s + 4·89-s + ⋯ |
| L(s) = 1 | + 2-s − 8/5·5-s − 8-s − 8/5·10-s − 3.38·13-s − 16-s + 1.29·17-s − 0.0799·25-s − 3.38·26-s − 2.34·29-s + 1.29·34-s − 0.864·37-s + 8/5·40-s − 0.634·41-s + 1.75·49-s − 0.0799·50-s − 1.96·53-s − 2.34·58-s − 0.524·61-s + 64-s + 5.41·65-s − 0.684·73-s − 0.864·74-s + 8/5·80-s − 0.634·82-s − 2.07·85-s + 4/89·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.05557332110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05557332110\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 86 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 95 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 479 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 470 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1874 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1175 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4406 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 52 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 4079 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11714 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12578 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 43 T + p^{2} T^{2} )^{2} \) |
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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
−12.16687437668462091104790842322, −11.44408582024080247574252321488, −10.91998139205902570350720362843, −10.02136107865034756329437121277, −9.907240436237270001320970052939, −9.340798525142447161276401112844, −8.897995856692441783734792942107, −8.004856452859394810806702823993, −7.72604143941003819461615891172, −7.33531777423026067996840947870, −7.04981515771174708110289281762, −6.07554952496415258037269657421, −5.34816475989614953072269974695, −5.17682551419295047492980108134, −4.49697842583467922159930384649, −4.00017328884712627468985694417, −3.46032327380077039916406823055, −2.85065314871626626255400877561, −2.01908646439889452572414761392, −0.089793227576284730007959020522,
0.089793227576284730007959020522, 2.01908646439889452572414761392, 2.85065314871626626255400877561, 3.46032327380077039916406823055, 4.00017328884712627468985694417, 4.49697842583467922159930384649, 5.17682551419295047492980108134, 5.34816475989614953072269974695, 6.07554952496415258037269657421, 7.04981515771174708110289281762, 7.33531777423026067996840947870, 7.72604143941003819461615891172, 8.004856452859394810806702823993, 8.897995856692441783734792942107, 9.340798525142447161276401112844, 9.907240436237270001320970052939, 10.02136107865034756329437121277, 10.91998139205902570350720362843, 11.44408582024080247574252321488, 12.16687437668462091104790842322
