| L(s) = 1 | + 3-s + 4-s − 5·7-s − 2·9-s + 12-s − 4·13-s − 3·16-s + 19-s − 5·21-s + 7·25-s − 5·27-s − 5·28-s − 14·31-s − 2·36-s − 2·37-s − 4·39-s − 12·43-s − 3·48-s + 5·49-s − 4·52-s + 57-s + 12·61-s + 10·63-s − 7·64-s + 23·67-s − 5·73-s + 7·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 1.88·7-s − 2/3·9-s + 0.288·12-s − 1.10·13-s − 3/4·16-s + 0.229·19-s − 1.09·21-s + 7/5·25-s − 0.962·27-s − 0.944·28-s − 2.51·31-s − 1/3·36-s − 0.328·37-s − 0.640·39-s − 1.82·43-s − 0.433·48-s + 5/7·49-s − 0.554·52-s + 0.132·57-s + 1.53·61-s + 1.25·63-s − 7/8·64-s + 2.80·67-s − 0.585·73-s + 0.808·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 6361 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 124 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 43 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 91 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 128 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 69 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−9.746631248281511248877325155199, −9.309497741758762994499318400415, −8.782024897965907841636861499858, −8.364091402035647217481857053327, −7.47520792151717437800393664904, −7.14871835272308105238171460492, −6.55535134041316075893889692379, −6.32710548034214412379274702935, −5.24728055883496052082717155627, −5.09299787403446468950255578264, −3.72609707128623372653467611045, −3.43221389911543551770704413581, −2.69541351369120047199854891228, −2.11569875822915840641086406730, 0,
2.11569875822915840641086406730, 2.69541351369120047199854891228, 3.43221389911543551770704413581, 3.72609707128623372653467611045, 5.09299787403446468950255578264, 5.24728055883496052082717155627, 6.32710548034214412379274702935, 6.55535134041316075893889692379, 7.14871835272308105238171460492, 7.47520792151717437800393664904, 8.364091402035647217481857053327, 8.782024897965907841636861499858, 9.309497741758762994499318400415, 9.746631248281511248877325155199
