| L(s) = 1 | + 2-s + 2·4-s − 3·5-s + 6·7-s + 5·8-s − 3·9-s − 3·10-s + 6·11-s − 10·13-s + 6·14-s + 5·16-s + 5·17-s − 3·18-s − 7·19-s − 6·20-s + 6·22-s + 2·23-s + 5·25-s − 10·26-s + 12·28-s − 2·29-s − 8·31-s + 10·32-s + 5·34-s − 18·35-s − 6·36-s + 9·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s − 1.34·5-s + 2.26·7-s + 1.76·8-s − 9-s − 0.948·10-s + 1.80·11-s − 2.77·13-s + 1.60·14-s + 5/4·16-s + 1.21·17-s − 0.707·18-s − 1.60·19-s − 1.34·20-s + 1.27·22-s + 0.417·23-s + 25-s − 1.96·26-s + 2.26·28-s − 0.371·29-s − 1.43·31-s + 1.76·32-s + 0.857·34-s − 3.04·35-s − 36-s + 1.47·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363609 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.490250903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.490250903\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 67 | $C_2$ | \( 1 + 11 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8 T - 19 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−11.05962738914622283471791875331, −10.62911992857046248062198529303, −10.38002840472889641549462147589, −9.319566831978486816526946791024, −9.291983296460589830129303957376, −8.391671576628489930011588244018, −8.105185571916260216741135596441, −7.55169348505908600901957596284, −7.49183963907470337622431061394, −7.11621318383934166135101082080, −6.39787956211372855671869373329, −5.73823317837946648153958496276, −5.10455485393123910196587784108, −4.80931595530805523102553255902, −4.19692455015729538347112986328, −4.16797969850106284201644296309, −3.22839536676073634865731056884, −2.24719398155894618166099548062, −2.05202057111786974588720892467, −0.992050466822613924866914908397,
0.992050466822613924866914908397, 2.05202057111786974588720892467, 2.24719398155894618166099548062, 3.22839536676073634865731056884, 4.16797969850106284201644296309, 4.19692455015729538347112986328, 4.80931595530805523102553255902, 5.10455485393123910196587784108, 5.73823317837946648153958496276, 6.39787956211372855671869373329, 7.11621318383934166135101082080, 7.49183963907470337622431061394, 7.55169348505908600901957596284, 8.105185571916260216741135596441, 8.391671576628489930011588244018, 9.291983296460589830129303957376, 9.319566831978486816526946791024, 10.38002840472889641549462147589, 10.62911992857046248062198529303, 11.05962738914622283471791875331
