| L(s) = 1 | + 4-s − 2·5-s + 3·7-s + 4·9-s − 10·13-s − 3·16-s − 2·20-s − 10·23-s − 25-s + 3·28-s − 4·29-s − 6·35-s + 4·36-s − 8·45-s − 10·52-s − 2·59-s + 12·63-s − 7·64-s + 20·65-s − 10·67-s + 6·80-s + 7·81-s − 20·83-s − 30·91-s − 10·92-s − 100-s + 10·103-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s + 1.13·7-s + 4/3·9-s − 2.77·13-s − 3/4·16-s − 0.447·20-s − 2.08·23-s − 1/5·25-s + 0.566·28-s − 0.742·29-s − 1.01·35-s + 2/3·36-s − 1.19·45-s − 1.38·52-s − 0.260·59-s + 1.51·63-s − 7/8·64-s + 2.48·65-s − 1.22·67-s + 0.670·80-s + 7/9·81-s − 2.19·83-s − 3.14·91-s − 1.04·92-s − 0.0999·100-s + 0.985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147175 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147175 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 154 T^{2} + p^{2} T^{4} \) |
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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
−9.128637484329168701980442083715, −8.387079494464761652684378565054, −7.88489983226529676624518311534, −7.50178636524173500825988650424, −7.32898041801012218373760230084, −6.86807730144545685027240607357, −6.09978428693942855628955826319, −5.40435942496967759536988150187, −4.74531326021182669140633075787, −4.39311344620761008007941681974, −4.08023284241150014931068510028, −3.02204521262610588733332733168, −2.08452236063289611265548006346, −1.84009393992423594341613556332, 0,
1.84009393992423594341613556332, 2.08452236063289611265548006346, 3.02204521262610588733332733168, 4.08023284241150014931068510028, 4.39311344620761008007941681974, 4.74531326021182669140633075787, 5.40435942496967759536988150187, 6.09978428693942855628955826319, 6.86807730144545685027240607357, 7.32898041801012218373760230084, 7.50178636524173500825988650424, 7.88489983226529676624518311534, 8.387079494464761652684378565054, 9.128637484329168701980442083715
