| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 4·9-s − 4·10-s − 4·11-s − 4·13-s + 5·16-s + 8·18-s − 4·19-s + 6·20-s + 8·22-s + 4·23-s + 3·25-s + 8·26-s − 8·31-s − 6·32-s − 12·36-s + 12·37-s + 8·38-s − 8·40-s − 12·44-s − 8·45-s − 8·46-s − 8·47-s − 6·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.41·8-s − 4/3·9-s − 1.26·10-s − 1.20·11-s − 1.10·13-s + 5/4·16-s + 1.88·18-s − 0.917·19-s + 1.34·20-s + 1.70·22-s + 0.834·23-s + 3/5·25-s + 1.56·26-s − 1.43·31-s − 1.06·32-s − 2·36-s + 1.97·37-s + 1.29·38-s − 1.26·40-s − 1.80·44-s − 1.19·45-s − 1.17·46-s − 1.16·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16080100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16080100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 401 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 28 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 92 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 100 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 24 T + 270 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 24 T + 336 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
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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
−8.262904094319191361507227107984, −8.201342374485404855248049227451, −7.52501091905095213792977054984, −7.21761558155977518386518425241, −6.97425091590272825847576907324, −6.58906575102368516064154302105, −5.95452393832042527191465372701, −5.76598780134220337763630466763, −5.39053380678540268544719861441, −5.22460400258152315103227763787, −4.38904211415536314128953945640, −4.22406675490055117954095991467, −3.08140407751815706824200618483, −3.05095241014274186966187772828, −2.54593829013625724921113772202, −2.26121726327558143396663957708, −1.70666197298114484740097066046, −1.05444077890884971116438749647, 0, 0,
1.05444077890884971116438749647, 1.70666197298114484740097066046, 2.26121726327558143396663957708, 2.54593829013625724921113772202, 3.05095241014274186966187772828, 3.08140407751815706824200618483, 4.22406675490055117954095991467, 4.38904211415536314128953945640, 5.22460400258152315103227763787, 5.39053380678540268544719861441, 5.76598780134220337763630466763, 5.95452393832042527191465372701, 6.58906575102368516064154302105, 6.97425091590272825847576907324, 7.21761558155977518386518425241, 7.52501091905095213792977054984, 8.201342374485404855248049227451, 8.262904094319191361507227107984
