| L(s) = 1 | − 3-s − 7-s − 9-s + 2·11-s + 3·13-s + 7·17-s − 6·19-s + 21-s − 2·23-s + 7·29-s + 2·31-s − 2·33-s − 2·37-s − 3·39-s − 12·41-s − 3·43-s − 12·47-s − 9·49-s − 7·51-s + 10·53-s + 6·57-s − 8·59-s − 12·61-s + 63-s + 7·67-s + 2·69-s + 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 1/3·9-s + 0.603·11-s + 0.832·13-s + 1.69·17-s − 1.37·19-s + 0.218·21-s − 0.417·23-s + 1.29·29-s + 0.359·31-s − 0.348·33-s − 0.328·37-s − 0.480·39-s − 1.87·41-s − 0.457·43-s − 1.75·47-s − 9/7·49-s − 0.980·51-s + 1.37·53-s + 0.794·57-s − 1.04·59-s − 1.53·61-s + 0.125·63-s + 0.855·67-s + 0.240·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_4$ | \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 142 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 134 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 133 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T - 48 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 153 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 284 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
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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
−7.45805055834973276415112160269, −7.16713834825860407270887220107, −6.70569435342780890354124176758, −6.46038430107011925615243600644, −6.12140244637848713431356986708, −6.09462505535339498007143195884, −5.40150038248284910150191119814, −5.27687932966146298568100798365, −4.80960700756797036913922381738, −4.41983261632235161774123662784, −4.03165442843343894093167900608, −3.60247979219192869265030069107, −3.18718823799585027484714742963, −3.08186573641713357111558218049, −2.41773900264012880943046415775, −1.83403424385472121079933898283, −1.32764384928443580921568721244, −1.14208435561055368084828906031, 0, 0,
1.14208435561055368084828906031, 1.32764384928443580921568721244, 1.83403424385472121079933898283, 2.41773900264012880943046415775, 3.08186573641713357111558218049, 3.18718823799585027484714742963, 3.60247979219192869265030069107, 4.03165442843343894093167900608, 4.41983261632235161774123662784, 4.80960700756797036913922381738, 5.27687932966146298568100798365, 5.40150038248284910150191119814, 6.09462505535339498007143195884, 6.12140244637848713431356986708, 6.46038430107011925615243600644, 6.70569435342780890354124176758, 7.16713834825860407270887220107, 7.45805055834973276415112160269
