| L(s) = 1 | + 2·4-s + 10·7-s − 2·13-s − 7·19-s + 5·25-s + 20·28-s + 22·31-s − 20·37-s − 5·43-s + 61·49-s − 4·52-s − 14·61-s − 8·64-s − 11·67-s − 17·73-s − 14·76-s + 4·79-s − 20·91-s + 19·97-s + 10·100-s − 14·103-s − 2·109-s − 22·121-s + 44·124-s + 127-s + 131-s − 70·133-s + ⋯ |
| L(s) = 1 | + 4-s + 3.77·7-s − 0.554·13-s − 1.60·19-s + 25-s + 3.77·28-s + 3.95·31-s − 3.28·37-s − 0.762·43-s + 61/7·49-s − 0.554·52-s − 1.79·61-s − 64-s − 1.34·67-s − 1.98·73-s − 1.60·76-s + 0.450·79-s − 2.09·91-s + 1.92·97-s + 100-s − 1.37·103-s − 0.191·109-s − 2·121-s + 3.95·124-s + 0.0887·127-s + 0.0873·131-s − 6.06·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 263169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 263169 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.508856107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.508856107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−11.11083399786877976108077511572, −10.58592505329486630887268076305, −10.39236798052945236492289683544, −10.35238039700419636036273596060, −8.906730718200671884712174980103, −8.863847752682789718666330898375, −8.214974859432309818884954752931, −8.164531753294053306435124703147, −7.55257414388467746803797631238, −7.14808467908576129792057564704, −6.57383711591087720070231036277, −6.14957824006156347044701679138, −5.14420049265614580957555023161, −5.07112364370699800649494382493, −4.38622861129775322387088944795, −4.36778427391294137867839513282, −2.93272057649844329678739917963, −2.37514158233360280319390967487, −1.71039201531404868381222688162, −1.38005630313448437283134518518,
1.38005630313448437283134518518, 1.71039201531404868381222688162, 2.37514158233360280319390967487, 2.93272057649844329678739917963, 4.36778427391294137867839513282, 4.38622861129775322387088944795, 5.07112364370699800649494382493, 5.14420049265614580957555023161, 6.14957824006156347044701679138, 6.57383711591087720070231036277, 7.14808467908576129792057564704, 7.55257414388467746803797631238, 8.164531753294053306435124703147, 8.214974859432309818884954752931, 8.863847752682789718666330898375, 8.906730718200671884712174980103, 10.35238039700419636036273596060, 10.39236798052945236492289683544, 10.58592505329486630887268076305, 11.11083399786877976108077511572
