| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s − 2·9-s + 5·16-s + 4·18-s − 10·25-s − 8·31-s − 6·32-s − 6·36-s + 4·37-s + 6·41-s + 16·43-s + 49-s + 20·50-s − 12·59-s + 16·61-s + 16·62-s + 7·64-s + 8·72-s + 4·73-s − 8·74-s − 5·81-s − 12·82-s − 12·83-s − 32·86-s − 2·98-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s − 2/3·9-s + 5/4·16-s + 0.942·18-s − 2·25-s − 1.43·31-s − 1.06·32-s − 36-s + 0.657·37-s + 0.937·41-s + 2.43·43-s + 1/7·49-s + 2.82·50-s − 1.56·59-s + 2.04·61-s + 2.03·62-s + 7/8·64-s + 0.942·72-s + 0.468·73-s − 0.929·74-s − 5/9·81-s − 1.32·82-s − 1.31·83-s − 3.45·86-s − 0.202·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 329476 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 329476 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 41 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.576574175927212899111454819532, −8.131298345473117072946549713145, −7.57571100088867902110310233811, −7.46852791716783421092039703831, −6.86761839836579218858889642687, −6.14364779323247904643646496119, −5.76672260141591023723183382906, −5.57928681742950427486583645839, −4.55033994430147192557970938084, −3.92908850742833290739203075499, −3.35438962358155565091884257335, −2.46869134005901036292907277411, −2.14361086660228166648215843038, −1.11551143424508389710750581105, 0,
1.11551143424508389710750581105, 2.14361086660228166648215843038, 2.46869134005901036292907277411, 3.35438962358155565091884257335, 3.92908850742833290739203075499, 4.55033994430147192557970938084, 5.57928681742950427486583645839, 5.76672260141591023723183382906, 6.14364779323247904643646496119, 6.86761839836579218858889642687, 7.46852791716783421092039703831, 7.57571100088867902110310233811, 8.131298345473117072946549713145, 8.576574175927212899111454819532
