| L(s) = 1 | − 2·3-s + 5·4-s − 6·5-s + 6·7-s + 2·9-s − 10·12-s + 12·15-s + 12·16-s + 12·17-s − 30·20-s − 12·21-s + 5·25-s − 6·27-s + 30·28-s − 36·35-s + 10·36-s + 32·37-s + 8·41-s + 26·43-s − 12·45-s + 32·47-s − 24·48-s + 18·49-s − 24·51-s − 10·59-s + 60·60-s + 12·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 5/2·4-s − 2.68·5-s + 2.26·7-s + 2/3·9-s − 2.88·12-s + 3.09·15-s + 3·16-s + 2.91·17-s − 6.70·20-s − 2.61·21-s + 25-s − 1.15·27-s + 5.66·28-s − 6.08·35-s + 5/3·36-s + 5.26·37-s + 1.24·41-s + 3.96·43-s − 1.78·45-s + 4.66·47-s − 3.46·48-s + 18/7·49-s − 3.36·51-s − 1.30·59-s + 7.74·60-s + 1.51·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.209996232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.209996232\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 13 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5 T^{2} - 207 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 49 p T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 74 T^{2} + 2731 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 211 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 16 T + 133 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 13 T + 127 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 + 15 T^{2} + 713 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 5 T + 23 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 209 T^{2} + 18081 T^{4} - 209 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 19 T + 223 T^{2} + 19 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 149 T^{2} + 14101 T^{4} - 149 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 210 T^{2} + 20963 T^{4} - 210 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 20 T + 253 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 137 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 130 T^{2} + 7363 T^{4} - 130 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81184580704691757893158130539, −7.78810662816369486143713918226, −7.39312133864889674666895858711, −7.38960518737465808985714979799, −7.26074160231560375929796028441, −7.09300225927295986012089836964, −6.24950225500305866874104666780, −6.03698500824422165967690122463, −5.86613417956197558177144932963, −5.76675511588821245096844928521, −5.64323863464469169906565641734, −5.61698431371832350317012482678, −4.64404557229709651744000741985, −4.40040607583075241343838917013, −4.24542615017828239053790774945, −4.19718093654593616556289346795, −4.13164799423517797233267763376, −3.29935330698309202818785611910, −3.18407411443623148358256842240, −2.60044587185931908194168802920, −2.49494862096002136280773891981, −2.12642090823846021802669485036, −1.41308452259816050191723358517, −1.07632849649512588977219094797, −0.798829303941988403988768756800,
0.798829303941988403988768756800, 1.07632849649512588977219094797, 1.41308452259816050191723358517, 2.12642090823846021802669485036, 2.49494862096002136280773891981, 2.60044587185931908194168802920, 3.18407411443623148358256842240, 3.29935330698309202818785611910, 4.13164799423517797233267763376, 4.19718093654593616556289346795, 4.24542615017828239053790774945, 4.40040607583075241343838917013, 4.64404557229709651744000741985, 5.61698431371832350317012482678, 5.64323863464469169906565641734, 5.76675511588821245096844928521, 5.86613417956197558177144932963, 6.03698500824422165967690122463, 6.24950225500305866874104666780, 7.09300225927295986012089836964, 7.26074160231560375929796028441, 7.38960518737465808985714979799, 7.39312133864889674666895858711, 7.78810662816369486143713918226, 7.81184580704691757893158130539
