| L(s) = 1 | + 8·7-s + 14·13-s + 16·19-s + 58·31-s − 4·37-s + 14·43-s − 50·49-s + 124·61-s + 116·67-s + 104·73-s − 98·79-s + 112·91-s + 68·97-s − 196·103-s + 52·109-s + 197·121-s + 127-s + 131-s + 128·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 191·169-s + ⋯ |
| L(s) = 1 | + 8/7·7-s + 1.07·13-s + 0.842·19-s + 1.87·31-s − 0.108·37-s + 0.325·43-s − 1.02·49-s + 2.03·61-s + 1.73·67-s + 1.42·73-s − 1.24·79-s + 1.23·91-s + 0.701·97-s − 1.90·103-s + 0.477·109-s + 1.62·121-s + 0.00787·127-s + 0.00763·131-s + 0.962·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.13·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.691761694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.691761694\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 197 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 173 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1013 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 523 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 29 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3182 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3293 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3202 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 5342 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 7202 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 52 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 49 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13598 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )^{2} \) |
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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
−8.565541279068201678278954973660, −8.434061798178957273520451819621, −8.111197071902940814943674820783, −7.996600339948938694877620482296, −7.23003461584255065513397150163, −7.04436689229243711310860950955, −6.58327540388732353701504103654, −6.12938205112103440322370164785, −5.79482697042153267451153792203, −5.28507619229985247104873169409, −4.94218561778789408062852164984, −4.64063401695141065317521571302, −3.92669484754689818062320109565, −3.87478849110491896472061736313, −2.96091346398945191896926698185, −2.91850542461819671961444090688, −1.86919755747770196568461948476, −1.82525451025387048596922460314, −0.808071054063721318869545294482, −0.791657995410884000049234679693,
0.791657995410884000049234679693, 0.808071054063721318869545294482, 1.82525451025387048596922460314, 1.86919755747770196568461948476, 2.91850542461819671961444090688, 2.96091346398945191896926698185, 3.87478849110491896472061736313, 3.92669484754689818062320109565, 4.64063401695141065317521571302, 4.94218561778789408062852164984, 5.28507619229985247104873169409, 5.79482697042153267451153792203, 6.12938205112103440322370164785, 6.58327540388732353701504103654, 7.04436689229243711310860950955, 7.23003461584255065513397150163, 7.996600339948938694877620482296, 8.111197071902940814943674820783, 8.434061798178957273520451819621, 8.565541279068201678278954973660
