| L(s) = 1 | − 3-s + 9-s + 4·11-s + 4·17-s − 6·25-s − 27-s + 12·29-s + 16·31-s − 4·33-s + 12·37-s − 12·41-s − 14·49-s − 4·51-s − 8·67-s + 6·75-s + 81-s − 8·83-s − 12·87-s − 16·93-s + 4·97-s + 4·99-s − 36·101-s + 32·103-s − 24·107-s − 12·111-s + 5·121-s + 12·123-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.970·17-s − 6/5·25-s − 0.192·27-s + 2.22·29-s + 2.87·31-s − 0.696·33-s + 1.97·37-s − 1.87·41-s − 2·49-s − 0.560·51-s − 0.977·67-s + 0.692·75-s + 1/9·81-s − 0.878·83-s − 1.28·87-s − 1.65·93-s + 0.406·97-s + 0.402·99-s − 3.58·101-s + 3.15·103-s − 2.32·107-s − 1.13·111-s + 5/11·121-s + 1.08·123-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.619117076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.619117076\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( 1 + T \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.166783448349530625953262482149, −8.412059248281827571082850459164, −8.098990694093691505068092710868, −7.83654723068629114933839597949, −6.92796334648786182474479663711, −6.42897107072744719896577758454, −6.40725479078387541676599821855, −5.70681982833013824339552087978, −5.05736853539339825666279063330, −4.38461660538881471089251752313, −4.25303028692796488061253338187, −3.21029387756548432515894957212, −2.80691305312846013255691010566, −1.61011393533291281462985460586, −0.918192672118257559483050015003,
0.918192672118257559483050015003, 1.61011393533291281462985460586, 2.80691305312846013255691010566, 3.21029387756548432515894957212, 4.25303028692796488061253338187, 4.38461660538881471089251752313, 5.05736853539339825666279063330, 5.70681982833013824339552087978, 6.40725479078387541676599821855, 6.42897107072744719896577758454, 6.92796334648786182474479663711, 7.83654723068629114933839597949, 8.098990694093691505068092710868, 8.412059248281827571082850459164, 9.166783448349530625953262482149
