| L(s) = 1 | + 2·2-s + 2·3-s − 4-s + 4·6-s − 8·8-s + 9-s − 2·12-s − 7·16-s − 10·17-s + 2·18-s − 16·24-s − 9·25-s − 4·27-s + 18·29-s − 4·31-s + 14·32-s − 20·34-s − 36-s − 6·37-s − 10·41-s − 14·48-s − 10·49-s − 18·50-s − 20·51-s − 8·54-s + 36·58-s − 8·62-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s − 1/2·4-s + 1.63·6-s − 2.82·8-s + 1/3·9-s − 0.577·12-s − 7/4·16-s − 2.42·17-s + 0.471·18-s − 3.26·24-s − 9/5·25-s − 0.769·27-s + 3.34·29-s − 0.718·31-s + 2.47·32-s − 3.42·34-s − 1/6·36-s − 0.986·37-s − 1.56·41-s − 2.02·48-s − 1.42·49-s − 2.54·50-s − 2.80·51-s − 1.08·54-s + 4.72·58-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
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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
−9.182457967238804819066801847611, −8.507285126107606377633981082529, −8.356936232923193217562617522792, −8.010979443598278861073897866124, −6.74272981252089439582006678154, −6.72221402891887535126565651102, −5.97336865191641765106303088244, −5.38415572352897256696127903868, −4.68607726153727393582211790246, −4.50122243823954701224397495329, −3.84177902362047353880486406829, −3.33556810952068997685484956976, −2.75274084463573300677827364197, −2.02313919782203708817074154348, 0,
2.02313919782203708817074154348, 2.75274084463573300677827364197, 3.33556810952068997685484956976, 3.84177902362047353880486406829, 4.50122243823954701224397495329, 4.68607726153727393582211790246, 5.38415572352897256696127903868, 5.97336865191641765106303088244, 6.72221402891887535126565651102, 6.74272981252089439582006678154, 8.010979443598278861073897866124, 8.356936232923193217562617522792, 8.507285126107606377633981082529, 9.182457967238804819066801847611
