L(s) = 1 | + 2-s − 4-s − 3·8-s − 9-s − 16-s − 18-s − 25-s + 5·32-s + 36-s − 7·49-s − 50-s + 7·64-s + 16·71-s + 3·72-s + 81-s − 7·98-s + 100-s − 4·113-s − 6·121-s + 127-s − 3·128-s + 131-s + 137-s + 139-s + 16·142-s + 144-s + 149-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s − 1/3·9-s − 1/4·16-s − 0.235·18-s − 1/5·25-s + 0.883·32-s + 1/6·36-s − 49-s − 0.141·50-s + 7/8·64-s + 1.89·71-s + 0.353·72-s + 1/9·81-s − 0.707·98-s + 1/10·100-s − 0.376·113-s − 0.545·121-s + 0.0887·127-s − 0.265·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.34·142-s + 1/12·144-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.106878665527196501488738029768, −7.77221046713237168946486645571, −7.11971522303416405292810269623, −6.54688851024595752035465721198, −6.29855002065771547155921166618, −5.66807838137471404469030056489, −5.31759258948085007443571553816, −4.84516250294896029701802728445, −4.38824987909813670639777395844, −3.75776204854414736459046056773, −3.42673070892602632805527335366, −2.75063685802091318252119195049, −2.16683693140602774553305713214, −1.13451403080775083022194990483, 0,
1.13451403080775083022194990483, 2.16683693140602774553305713214, 2.75063685802091318252119195049, 3.42673070892602632805527335366, 3.75776204854414736459046056773, 4.38824987909813670639777395844, 4.84516250294896029701802728445, 5.31759258948085007443571553816, 5.66807838137471404469030056489, 6.29855002065771547155921166618, 6.54688851024595752035465721198, 7.11971522303416405292810269623, 7.77221046713237168946486645571, 8.106878665527196501488738029768