| L(s) = 1 | + 4·25-s + 2·49-s + 4·73-s − 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4·25-s + 2·49-s + 4·73-s − 4·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.421238640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421238640\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
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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
−6.91502407249148196331712360437, −6.59155415689068975832667498616, −6.38398587584387894507570288870, −6.26291753977632947805453132930, −6.23812807332223789270393263036, −5.65695255260783265523115516377, −5.38873634716442405665106788222, −5.31772185895697473213617645467, −5.21838026674552820286354839953, −4.97030408834250382418989449407, −4.72147959048422890313670555702, −4.34475742944151986063552940965, −4.33032740869649264700704894474, −3.91197042016979219128012983439, −3.77713839038321684515577177177, −3.45381103216667111422212066484, −3.24892272853930578038120055939, −2.89939982696923096856465047296, −2.60489454967546193983758697352, −2.46821184475163916363422725550, −2.36954644930763375406923510681, −1.77589898318638160448823422527, −1.28671355777553037341701740716, −1.15771038397960263291517791310, −0.78014142159824132685990337536,
0.78014142159824132685990337536, 1.15771038397960263291517791310, 1.28671355777553037341701740716, 1.77589898318638160448823422527, 2.36954644930763375406923510681, 2.46821184475163916363422725550, 2.60489454967546193983758697352, 2.89939982696923096856465047296, 3.24892272853930578038120055939, 3.45381103216667111422212066484, 3.77713839038321684515577177177, 3.91197042016979219128012983439, 4.33032740869649264700704894474, 4.34475742944151986063552940965, 4.72147959048422890313670555702, 4.97030408834250382418989449407, 5.21838026674552820286354839953, 5.31772185895697473213617645467, 5.38873634716442405665106788222, 5.65695255260783265523115516377, 6.23812807332223789270393263036, 6.26291753977632947805453132930, 6.38398587584387894507570288870, 6.59155415689068975832667498616, 6.91502407249148196331712360437
