| L(s) = 1 | − 2·4-s − 2·9-s + 4·11-s + 16-s + 25-s + 4·29-s + 4·36-s − 8·44-s − 2·49-s + 2·64-s + 3·81-s − 8·99-s − 2·100-s − 8·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2·4-s − 2·9-s + 4·11-s + 16-s + 25-s + 4·29-s + 4·36-s − 8·44-s − 2·49-s + 2·64-s + 3·81-s − 8·99-s − 2·100-s − 8·116-s + 10·121-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6691240997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6691240997\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−7.06336513544506538569940177907, −6.84908906968068297899502260078, −6.82629250369933062345625548732, −6.35547569838516215597372815270, −6.32099296482791001755868260594, −6.17744550234649468148045970213, −6.16889981728502958607626062766, −5.63142671380701312295180660042, −5.30049247066631226973566099998, −4.99174044644371470207611855904, −4.98959595128720236955227606167, −4.66475066800720105042113560230, −4.55061992189884802013169431075, −4.13029624276858724731007393205, −4.09896801006287131778281496356, −3.82067375058755751709208077572, −3.54365848832387636983011283052, −3.16938837403041654713167136728, −2.91894950487452125863665935181, −2.91370265807162717433781706912, −2.26491839036887251555052154813, −1.96201075347461263757079181143, −1.41135047742484393680961276344, −0.995465327582597890749757200938, −0.838687905402583285951691983613,
0.838687905402583285951691983613, 0.995465327582597890749757200938, 1.41135047742484393680961276344, 1.96201075347461263757079181143, 2.26491839036887251555052154813, 2.91370265807162717433781706912, 2.91894950487452125863665935181, 3.16938837403041654713167136728, 3.54365848832387636983011283052, 3.82067375058755751709208077572, 4.09896801006287131778281496356, 4.13029624276858724731007393205, 4.55061992189884802013169431075, 4.66475066800720105042113560230, 4.98959595128720236955227606167, 4.99174044644371470207611855904, 5.30049247066631226973566099998, 5.63142671380701312295180660042, 6.16889981728502958607626062766, 6.17744550234649468148045970213, 6.32099296482791001755868260594, 6.35547569838516215597372815270, 6.82629250369933062345625548732, 6.84908906968068297899502260078, 7.06336513544506538569940177907
