| L(s) = 1 | − 2·2-s + 4·4-s − 6·5-s − 16·8-s + 12·10-s + 10·11-s + 32·16-s − 12·17-s − 57·19-s − 24·20-s − 20·22-s + 40·23-s − 25-s − 32·29-s − 9·31-s − 64·32-s + 24·34-s − 5·37-s + 114·38-s + 96·40-s − 38·43-s + 40·44-s − 80·46-s − 90·47-s + 2·50-s − 32·53-s − 60·55-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 6/5·5-s − 2·8-s + 6/5·10-s + 0.909·11-s + 2·16-s − 0.705·17-s − 3·19-s − 6/5·20-s − 0.909·22-s + 1.73·23-s − 0.0399·25-s − 1.10·29-s − 0.290·31-s − 2·32-s + 0.705·34-s − 0.135·37-s + 3·38-s + 12/5·40-s − 0.883·43-s + 0.909·44-s − 1.73·46-s − 1.91·47-s + 1/25·50-s − 0.603·53-s − 1.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.02110833474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02110833474\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 - p T + p^{2} T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 37 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T - 21 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )( 1 + 23 T + p^{2} T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 12 T + 337 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p T + p^{2} T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 40 T + 1071 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 9 T + 988 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 5 T - 1344 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2774 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 19 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 90 T + 4909 T^{2} + 90 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 32 T - 1785 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 72 T + 5209 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 36 T + 4153 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 59 T - 1008 T^{2} + 59 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 33 T + 5692 T^{2} - 33 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 47 T - 4032 T^{2} + 47 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13190 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 204 T + 21793 T^{2} - 204 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 16466 T^{2} + p^{4} T^{4} \) |
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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
−11.15113367044896144630225301419, −10.68585217274660652765375481896, −10.41611503576526807176591180634, −9.453093181243588158502859009290, −9.368290439030576737296390208329, −8.736875092662036178865871581038, −8.596106485280482062178582718888, −8.020343053389209018902035555334, −7.65036877750206653190365449827, −6.76876386378459087214517260639, −6.59118775819864231547453026177, −6.42096421267134793961322247443, −5.50839543987090264374933066922, −4.86202951587262594741707875807, −3.95367866718128500311744073584, −3.81526281066797401242702096295, −2.97746738063223539235820776962, −2.26945717283371648285257671497, −1.47351744087576657832303508520, −0.07118196991700430910186429155,
0.07118196991700430910186429155, 1.47351744087576657832303508520, 2.26945717283371648285257671497, 2.97746738063223539235820776962, 3.81526281066797401242702096295, 3.95367866718128500311744073584, 4.86202951587262594741707875807, 5.50839543987090264374933066922, 6.42096421267134793961322247443, 6.59118775819864231547453026177, 6.76876386378459087214517260639, 7.65036877750206653190365449827, 8.020343053389209018902035555334, 8.596106485280482062178582718888, 8.736875092662036178865871581038, 9.368290439030576737296390208329, 9.453093181243588158502859009290, 10.41611503576526807176591180634, 10.68585217274660652765375481896, 11.15113367044896144630225301419
