L(s) = 1 | + 4-s + 2·7-s − 2·11-s + 16-s − 12·23-s − 2·25-s + 2·28-s + 10·29-s + 2·37-s − 4·43-s − 2·44-s − 3·49-s − 10·53-s + 64-s + 2·67-s + 2·71-s − 4·77-s + 32·79-s − 12·92-s − 2·100-s − 2·107-s + 32·109-s + 2·112-s + 8·113-s + 10·116-s − 7·121-s + 127-s + ⋯ |
L(s) = 1 | + 1/2·4-s + 0.755·7-s − 0.603·11-s + 1/4·16-s − 2.50·23-s − 2/5·25-s + 0.377·28-s + 1.85·29-s + 0.328·37-s − 0.609·43-s − 0.301·44-s − 3/7·49-s − 1.37·53-s + 1/8·64-s + 0.244·67-s + 0.237·71-s − 0.455·77-s + 3.60·79-s − 1.25·92-s − 1/5·100-s − 0.193·107-s + 3.06·109-s + 0.188·112-s + 0.752·113-s + 0.928·116-s − 0.636·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.229530679\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.229530679\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
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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
−7.81075436877772422688307556323, −7.62843051817236631930650323185, −6.81296688451494752207752100713, −6.44663251048134589009182455155, −6.18277365331418528292080509274, −5.66866632344199800564023446950, −5.15799546261721427143191540474, −4.66806160201978787447096229457, −4.39943681389495218625336141302, −3.64219298398960059908347342207, −3.28950542305954212942286202865, −2.50968229729405733855511869633, −2.08825436589523074736601345935, −1.60416866702826461383835580268, −0.59638390898364198478340180300,
0.59638390898364198478340180300, 1.60416866702826461383835580268, 2.08825436589523074736601345935, 2.50968229729405733855511869633, 3.28950542305954212942286202865, 3.64219298398960059908347342207, 4.39943681389495218625336141302, 4.66806160201978787447096229457, 5.15799546261721427143191540474, 5.66866632344199800564023446950, 6.18277365331418528292080509274, 6.44663251048134589009182455155, 6.81296688451494752207752100713, 7.62843051817236631930650323185, 7.81075436877772422688307556323