L(s) = 1 | + 3-s − 4·4-s − 7-s − 2·9-s − 4·12-s + 12·16-s + 12·17-s − 21-s − 10·25-s − 5·27-s + 4·28-s + 8·36-s + 2·37-s − 18·41-s + 16·43-s + 6·47-s + 12·48-s − 6·49-s + 12·51-s + 24·59-s + 2·63-s − 32·64-s − 8·67-s − 48·68-s − 10·75-s − 20·79-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 2·4-s − 0.377·7-s − 2/3·9-s − 1.15·12-s + 3·16-s + 2.91·17-s − 0.218·21-s − 2·25-s − 0.962·27-s + 0.755·28-s + 4/3·36-s + 0.328·37-s − 2.81·41-s + 2.43·43-s + 0.875·47-s + 1.73·48-s − 6/7·49-s + 1.68·51-s + 3.12·59-s + 0.251·63-s − 4·64-s − 0.977·67-s − 5.82·68-s − 1.15·75-s − 2.25·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 603729 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.034247407\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.034247407\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.347987828239931604977367931358, −8.089215632520962089364900419950, −7.59911177067371416247669368567, −7.51700863182857444916648212116, −6.46976232199214034486269724910, −5.81190721532545578243846792642, −5.44973416215471530225317007593, −5.43600402261952430245712783789, −4.58763012435405230199591087932, −3.94859068116611992371513252418, −3.50910294340479626549928321873, −3.38651949790641488199704187423, −2.50241497573322457204613884521, −1.42406061381510793894101910809, −0.55251621019845393128929739547,
0.55251621019845393128929739547, 1.42406061381510793894101910809, 2.50241497573322457204613884521, 3.38651949790641488199704187423, 3.50910294340479626549928321873, 3.94859068116611992371513252418, 4.58763012435405230199591087932, 5.43600402261952430245712783789, 5.44973416215471530225317007593, 5.81190721532545578243846792642, 6.46976232199214034486269724910, 7.51700863182857444916648212116, 7.59911177067371416247669368567, 8.089215632520962089364900419950, 8.347987828239931604977367931358