| L(s) = 1 | + 3·7-s + 9-s + 4·17-s + 8·23-s + 25-s − 31-s − 10·41-s − 5·47-s + 5·49-s + 3·63-s + 9·71-s − 3·73-s − 15·79-s − 8·81-s + 4·89-s − 97-s + 25·103-s − 6·113-s + 12·119-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·153-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 1/3·9-s + 0.970·17-s + 1.66·23-s + 1/5·25-s − 0.179·31-s − 1.56·41-s − 0.729·47-s + 5/7·49-s + 0.377·63-s + 1.06·71-s − 0.351·73-s − 1.68·79-s − 8/9·81-s + 0.423·89-s − 0.101·97-s + 2.46·103-s − 0.564·113-s + 1.10·119-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.323·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104704 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.912172191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.912172191\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 409 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 16 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−9.515520283562581210973448688276, −8.977211971531894641632563031643, −8.546321518725811110579065025009, −8.080438596290399817216119305862, −7.57503688392736477220685905297, −7.08037394421531893882104409602, −6.66572981169686125466251145919, −5.86481147445301147713073337177, −5.27065823458196778096354009156, −4.91205930042621764394205094628, −4.34836418611443265584197878991, −3.49930095186891178761430078943, −2.94438912430494208566519812883, −1.89101265682560927897063313121, −1.14563604386588753233484073735,
1.14563604386588753233484073735, 1.89101265682560927897063313121, 2.94438912430494208566519812883, 3.49930095186891178761430078943, 4.34836418611443265584197878991, 4.91205930042621764394205094628, 5.27065823458196778096354009156, 5.86481147445301147713073337177, 6.66572981169686125466251145919, 7.08037394421531893882104409602, 7.57503688392736477220685905297, 8.080438596290399817216119305862, 8.546321518725811110579065025009, 8.977211971531894641632563031643, 9.515520283562581210973448688276
