| L(s) = 1 | − 2·4-s + 10·13-s − 19-s + 25-s + 3·31-s − 12·37-s − 10·43-s − 10·49-s − 20·52-s + 2·61-s + 8·64-s + 4·67-s − 8·73-s + 2·76-s + 2·79-s − 22·97-s − 2·100-s − 36·103-s − 7·109-s − 9·121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s + 24·148-s + 149-s + ⋯ |
| L(s) = 1 | − 4-s + 2.77·13-s − 0.229·19-s + 1/5·25-s + 0.538·31-s − 1.97·37-s − 1.52·43-s − 1.42·49-s − 2.77·52-s + 0.256·61-s + 64-s + 0.488·67-s − 0.936·73-s + 0.229·76-s + 0.225·79-s − 2.23·97-s − 1/5·100-s − 3.54·103-s − 0.670·109-s − 0.818·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.97·148-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 93 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.270731226826548888974275331436, −7.979861869097349205620026627173, −6.87244029092208968082688570656, −6.80574947248602853190266223077, −6.35450331544968665653471087535, −5.65924042232185999500062358436, −5.41049972917035471336152487853, −4.77929711168029645016524439593, −4.28187877298872610960741606633, −3.71984530305062655813366610869, −3.48894881547595653665387029712, −2.76768644633230574350606151134, −1.67643204003326032024474779265, −1.23589929378979944378282330870, 0,
1.23589929378979944378282330870, 1.67643204003326032024474779265, 2.76768644633230574350606151134, 3.48894881547595653665387029712, 3.71984530305062655813366610869, 4.28187877298872610960741606633, 4.77929711168029645016524439593, 5.41049972917035471336152487853, 5.65924042232185999500062358436, 6.35450331544968665653471087535, 6.80574947248602853190266223077, 6.87244029092208968082688570656, 7.979861869097349205620026627173, 8.270731226826548888974275331436
