| L(s) = 1 | − 14·5-s − 22·7-s + 6·11-s + 34·17-s + 38·19-s + 4·23-s + 97·25-s + 308·35-s + 42·43-s − 10·47-s + 265·49-s − 84·55-s + 46·61-s + 78·73-s − 132·77-s − 12·83-s − 476·85-s − 532·95-s − 244·101-s − 56·115-s − 748·119-s − 215·121-s − 322·125-s + 127-s + 131-s − 836·133-s + 137-s + ⋯ |
| L(s) = 1 | − 2.79·5-s − 3.14·7-s + 6/11·11-s + 2·17-s + 2·19-s + 4/23·23-s + 3.87·25-s + 44/5·35-s + 0.976·43-s − 0.212·47-s + 5.40·49-s − 1.52·55-s + 0.754·61-s + 1.06·73-s − 1.71·77-s − 0.144·83-s − 5.59·85-s − 5.59·95-s − 2.41·101-s − 0.486·115-s − 6.28·119-s − 1.77·121-s − 2.57·125-s + 0.00787·127-s + 0.00763·131-s − 6.28·133-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02657352400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02657352400\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 210 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 114 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 1890 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1170 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 1794 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 21 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 5 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 5586 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5810 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 7410 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 1890 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 39 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 3234 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1730 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )( 1 + 94 T + p^{2} T^{2} ) \) |
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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
−9.177305583376216898726970618883, −8.208850659089930954135162480977, −8.081135826890479297695974816702, −7.59151813140525724586077676305, −7.35407086765823846071304169297, −7.04680744191219891712001137407, −6.72442401389623998453843864096, −6.20141626398805930106019693827, −5.91529865222587186567521362696, −5.24965394737118642965173912021, −5.05983668902382265487330444527, −4.03188017027254194821607328455, −3.96103636500002537358321257262, −3.51711908111293772617283926915, −3.45799986699934229368587296129, −2.83421843604645816279824663538, −2.73915680265906315362163159474, −1.05039645358483421031747458107, −0.967490205696825415777589730866, −0.05492748429264239096485275246,
0.05492748429264239096485275246, 0.967490205696825415777589730866, 1.05039645358483421031747458107, 2.73915680265906315362163159474, 2.83421843604645816279824663538, 3.45799986699934229368587296129, 3.51711908111293772617283926915, 3.96103636500002537358321257262, 4.03188017027254194821607328455, 5.05983668902382265487330444527, 5.24965394737118642965173912021, 5.91529865222587186567521362696, 6.20141626398805930106019693827, 6.72442401389623998453843864096, 7.04680744191219891712001137407, 7.35407086765823846071304169297, 7.59151813140525724586077676305, 8.081135826890479297695974816702, 8.208850659089930954135162480977, 9.177305583376216898726970618883
