| L(s) = 1 | − 2·7-s + 4·13-s − 14·19-s − 2·31-s − 14·37-s + 34·43-s − 95·49-s − 86·61-s − 44·67-s + 94·73-s − 98·79-s − 8·91-s − 242·97-s + 214·103-s − 278·109-s + 62·121-s + 127-s + 131-s + 28·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 326·169-s + ⋯ |
| L(s) = 1 | − 2/7·7-s + 4/13·13-s − 0.736·19-s − 0.0645·31-s − 0.378·37-s + 0.790·43-s − 1.93·49-s − 1.40·61-s − 0.656·67-s + 1.28·73-s − 1.24·79-s − 0.0879·91-s − 2.49·97-s + 2.07·103-s − 2.55·109-s + 0.512·121-s + 0.00787·127-s + 0.00763·131-s + 4/19·133-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 − 1.92·169-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2519156630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2519156630\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 62 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 142 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )( 1 + 44 T + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 962 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1742 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5438 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6782 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 43 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8462 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 49 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13058 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1262 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 121 T + p^{2} T^{2} )^{2} \) |
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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
−8.917278750397641717451897117100, −8.386483020702183134326766322751, −8.198688669471368325422305264859, −7.65223055928857337346280818921, −7.46652692936361828285939946045, −6.77368073529367437145238923481, −6.66837909071271742749310393886, −6.05243560727510142200475898886, −6.01076915260021786531013597371, −5.20438490380602246758907215049, −5.11279218669461817080259970084, −4.31632962449046811813119557449, −4.27135135534846983661453229382, −3.52806133621856937061226097115, −3.28889274458272542671631245710, −2.61177653478821224178785575674, −2.29289030322444914479629994295, −1.48171475943200030156289721972, −1.20574080497244370752311615692, −0.11870043396517160932755271346,
0.11870043396517160932755271346, 1.20574080497244370752311615692, 1.48171475943200030156289721972, 2.29289030322444914479629994295, 2.61177653478821224178785575674, 3.28889274458272542671631245710, 3.52806133621856937061226097115, 4.27135135534846983661453229382, 4.31632962449046811813119557449, 5.11279218669461817080259970084, 5.20438490380602246758907215049, 6.01076915260021786531013597371, 6.05243560727510142200475898886, 6.66837909071271742749310393886, 6.77368073529367437145238923481, 7.46652692936361828285939946045, 7.65223055928857337346280818921, 8.198688669471368325422305264859, 8.386483020702183134326766322751, 8.917278750397641717451897117100
