L(s) = 1 | − 2·2-s − 3-s + 3·4-s + 5-s + 2·6-s − 4·8-s + 3·9-s − 2·10-s − 3·11-s − 3·12-s + 5·13-s − 15-s + 5·16-s − 2·17-s − 6·18-s + 5·19-s + 3·20-s + 6·22-s − 6·23-s + 4·24-s − 25-s − 10·26-s − 8·27-s + 3·29-s + 2·30-s + 6·31-s − 6·32-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.447·5-s + 0.816·6-s − 1.41·8-s + 9-s − 0.632·10-s − 0.904·11-s − 0.866·12-s + 1.38·13-s − 0.258·15-s + 5/4·16-s − 0.485·17-s − 1.41·18-s + 1.14·19-s + 0.670·20-s + 1.27·22-s − 1.25·23-s + 0.816·24-s − 1/5·25-s − 1.96·26-s − 1.53·27-s + 0.557·29-s + 0.365·30-s + 1.07·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 90 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 116 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 144 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 145 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 92 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 15 T + 206 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5 T + 164 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−8.215065317731750669523712556348, −7.84550287854155103548981645388, −7.80569613455061808977238691558, −7.21762784210767678889474145165, −6.84130161128538060594575949367, −6.42059711437723404237460200608, −6.20033011512328239835601118148, −5.73180706934821513024584011981, −5.61757763343157302413636243051, −4.79544973374189617563506639804, −4.62284262770275185999294553372, −4.10189178397253841740231205102, −3.39894159064024539565350543420, −3.05432970008752690765672867085, −2.70733686786480469302803442569, −1.75867203670746081516867686957, −1.45955029600441709954699210674, −1.40249507977420917152770130666, 0, 0,
1.40249507977420917152770130666, 1.45955029600441709954699210674, 1.75867203670746081516867686957, 2.70733686786480469302803442569, 3.05432970008752690765672867085, 3.39894159064024539565350543420, 4.10189178397253841740231205102, 4.62284262770275185999294553372, 4.79544973374189617563506639804, 5.61757763343157302413636243051, 5.73180706934821513024584011981, 6.20033011512328239835601118148, 6.42059711437723404237460200608, 6.84130161128538060594575949367, 7.21762784210767678889474145165, 7.80569613455061808977238691558, 7.84550287854155103548981645388, 8.215065317731750669523712556348