L(s) = 1 | + 2·3-s − 4-s + 4·5-s − 4·7-s + 2·9-s − 2·11-s − 2·12-s + 4·13-s + 8·15-s + 16-s − 2·17-s − 6·19-s − 4·20-s − 8·21-s + 2·23-s + 11·25-s + 6·27-s + 4·28-s + 2·31-s − 4·33-s − 16·35-s − 2·36-s + 16·37-s + 8·39-s − 14·41-s + 2·43-s + 2·44-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1/2·4-s + 1.78·5-s − 1.51·7-s + 2/3·9-s − 0.603·11-s − 0.577·12-s + 1.10·13-s + 2.06·15-s + 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.894·20-s − 1.74·21-s + 0.417·23-s + 11/5·25-s + 1.15·27-s + 0.755·28-s + 0.359·31-s − 0.696·33-s − 2.70·35-s − 1/3·36-s + 2.63·37-s + 1.28·39-s − 2.18·41-s + 0.304·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.605177765\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.605177765\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−13.64337110530095432106357131999, −13.05459281641719259178489730271, −12.92358644331471854043348906361, −12.59885894662536654305191701361, −11.35548037414773873533430140691, −10.82998032523111201228315194521, −10.03163407259311614154419274879, −9.988714052225930792873778562481, −9.396034766172495879402545659133, −8.925371997439647766362272555654, −8.436428541187349539410498002341, −8.002957984633436240618410672943, −6.74498304168338119841727422465, −6.42182849287379615300516084221, −6.08369063518413757126183913516, −5.07529002721091630239953100297, −4.36437902718636375213579142766, −3.17974475417872030817262495352, −2.88718171924783037467530224203, −1.78487513909741876656119990510,
1.78487513909741876656119990510, 2.88718171924783037467530224203, 3.17974475417872030817262495352, 4.36437902718636375213579142766, 5.07529002721091630239953100297, 6.08369063518413757126183913516, 6.42182849287379615300516084221, 6.74498304168338119841727422465, 8.002957984633436240618410672943, 8.436428541187349539410498002341, 8.925371997439647766362272555654, 9.396034766172495879402545659133, 9.988714052225930792873778562481, 10.03163407259311614154419274879, 10.82998032523111201228315194521, 11.35548037414773873533430140691, 12.59885894662536654305191701361, 12.92358644331471854043348906361, 13.05459281641719259178489730271, 13.64337110530095432106357131999