| L(s) = 1 | + 4·7-s − 8·25-s − 8·37-s − 16·43-s − 2·49-s − 32·67-s + 40·79-s − 9·81-s + 40·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s − 32·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 8/5·25-s − 1.31·37-s − 2.43·43-s − 2/7·49-s − 3.90·67-s + 4.50·79-s − 81-s + 3.83·109-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 0.0760·173-s − 2.41·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.881849603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.881849603\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.426063234914175552557143546419, −7.938365505532690691356189839986, −7.84871572056237082647927920266, −7.81058150602872162967003352215, −7.62705283706085861903720060917, −7.05267284069483291157687083712, −6.90414700101438436501067259618, −6.55128237827114139699749106896, −6.51285685121673954177423338479, −5.91120795892137436695666300086, −5.80875260929584413150654407164, −5.50645077544372047248198777221, −5.20608877605035818750003235730, −4.94668590781300491066236269412, −4.55861039427786597827888222785, −4.40753613167286934145771029347, −4.28279746559816202868099994921, −3.45656050005144128874871781220, −3.37876141762368388962096398358, −3.28162586125802272383561467627, −2.59495993088035092259271760076, −1.87172257433213335813484438276, −1.84416854678432403621113722917, −1.65291751012374252963680031852, −0.57760659627357354396129654729,
0.57760659627357354396129654729, 1.65291751012374252963680031852, 1.84416854678432403621113722917, 1.87172257433213335813484438276, 2.59495993088035092259271760076, 3.28162586125802272383561467627, 3.37876141762368388962096398358, 3.45656050005144128874871781220, 4.28279746559816202868099994921, 4.40753613167286934145771029347, 4.55861039427786597827888222785, 4.94668590781300491066236269412, 5.20608877605035818750003235730, 5.50645077544372047248198777221, 5.80875260929584413150654407164, 5.91120795892137436695666300086, 6.51285685121673954177423338479, 6.55128237827114139699749106896, 6.90414700101438436501067259618, 7.05267284069483291157687083712, 7.62705283706085861903720060917, 7.81058150602872162967003352215, 7.84871572056237082647927920266, 7.938365505532690691356189839986, 8.426063234914175552557143546419
