| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 4·8-s + 9-s − 4·10-s − 6·12-s + 4·13-s − 4·15-s + 5·16-s − 2·18-s + 6·20-s + 8·24-s + 3·25-s − 8·26-s + 4·27-s + 8·30-s + 16·31-s − 6·32-s + 3·36-s − 8·39-s − 8·40-s − 10·43-s + 2·45-s − 10·48-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 1.41·8-s + 1/3·9-s − 1.26·10-s − 1.73·12-s + 1.10·13-s − 1.03·15-s + 5/4·16-s − 0.471·18-s + 1.34·20-s + 1.63·24-s + 3/5·25-s − 1.56·26-s + 0.769·27-s + 1.46·30-s + 2.87·31-s − 1.06·32-s + 1/2·36-s − 1.28·39-s − 1.26·40-s − 1.52·43-s + 0.298·45-s − 1.44·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.048976860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048976860\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 43 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.744978262760076064470350841200, −9.681596380519631999091317973609, −9.100935464666348218227007063632, −8.541499794443180017685844823202, −8.292290700983134768028844357246, −8.199698856694699711853030151040, −7.17026411982525466619911325728, −7.13232332715327810297255231266, −6.39991706450826075644964834040, −6.30022298297411374670551802188, −5.88174840750081593045100454091, −5.54258962617628446686703270400, −4.69612269310507504694135080676, −4.68466548315348438310295870800, −3.55835749346130759474062513077, −3.17562360467250298420246166940, −2.37122968882016354872675325030, −1.96290525330784339879246940315, −0.976456061729641315701431144910, −0.77571786629065627069080704003,
0.77571786629065627069080704003, 0.976456061729641315701431144910, 1.96290525330784339879246940315, 2.37122968882016354872675325030, 3.17562360467250298420246166940, 3.55835749346130759474062513077, 4.68466548315348438310295870800, 4.69612269310507504694135080676, 5.54258962617628446686703270400, 5.88174840750081593045100454091, 6.30022298297411374670551802188, 6.39991706450826075644964834040, 7.13232332715327810297255231266, 7.17026411982525466619911325728, 8.199698856694699711853030151040, 8.292290700983134768028844357246, 8.541499794443180017685844823202, 9.100935464666348218227007063632, 9.681596380519631999091317973609, 9.744978262760076064470350841200
