| L(s) = 1 | − 2-s − 4·5-s + 7-s + 8-s + 4·10-s − 11-s + 5·13-s − 14-s − 16-s − 17-s − 19-s + 22-s − 6·23-s + 2·25-s − 5·26-s + 9·29-s + 4·31-s + 34-s − 4·35-s + 2·37-s + 38-s − 4·40-s + 5·41-s + 2·43-s + 6·46-s + 14·47-s − 2·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.78·5-s + 0.377·7-s + 0.353·8-s + 1.26·10-s − 0.301·11-s + 1.38·13-s − 0.267·14-s − 1/4·16-s − 0.242·17-s − 0.229·19-s + 0.213·22-s − 1.25·23-s + 2/5·25-s − 0.980·26-s + 1.67·29-s + 0.718·31-s + 0.171·34-s − 0.676·35-s + 0.328·37-s + 0.162·38-s − 0.632·40-s + 0.780·41-s + 0.304·43-s + 0.884·46-s + 2.04·47-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.171979323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.171979323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 11 T + 32 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.292331547125988709229220035373, −9.281586862059777053076041559965, −8.461937801810093315838870066978, −8.397721212748659819299683352104, −8.071596324692787540959118244146, −7.85090956410410088838018603094, −7.31632140133700598944019068931, −7.01588321119549711245770983563, −6.24538807820114956885930497354, −6.18751792445044470737282286500, −5.54960330106361078525678287487, −4.88739693085811555380308408318, −4.46955372551676031008136931411, −4.08560640460018635467507186504, −3.67873435797707428209209278588, −3.39713899259027853464851775936, −2.34557951609499445988554119705, −2.13025891806986356332369763752, −0.805364916281656454564745659634, −0.73145769133286488897206232803,
0.73145769133286488897206232803, 0.805364916281656454564745659634, 2.13025891806986356332369763752, 2.34557951609499445988554119705, 3.39713899259027853464851775936, 3.67873435797707428209209278588, 4.08560640460018635467507186504, 4.46955372551676031008136931411, 4.88739693085811555380308408318, 5.54960330106361078525678287487, 6.18751792445044470737282286500, 6.24538807820114956885930497354, 7.01588321119549711245770983563, 7.31632140133700598944019068931, 7.85090956410410088838018603094, 8.071596324692787540959118244146, 8.397721212748659819299683352104, 8.461937801810093315838870066978, 9.281586862059777053076041559965, 9.292331547125988709229220035373
