| L(s) = 1 | + 4·25-s − 4·49-s − 81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 4·25-s − 4·49-s − 81-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-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 + 239-s + 241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.143649952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.143649952\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 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
−6.89078908662822911633680426017, −6.69497054129257655538977364344, −6.51550777352264895345966044221, −6.40577597314999275202493327753, −6.26068990860901136944665667563, −5.84504045857925779397450540147, −5.58625318363115348573890864929, −5.34020912141235155753531105717, −5.32969437251898836552495153547, −4.75180123181174105061516764535, −4.68165531553386201126335297889, −4.66686379699381178383873813382, −4.57671531659987929678886950681, −4.04833588597752672251228191017, −3.61448118192798378442395823001, −3.53470374117965125402303784524, −3.25472605722643107306598150096, −3.12063464619350709198087589378, −2.68248220446785032416315016012, −2.51335733051674113947713163323, −2.38832022748634222207639775930, −1.60979645602740736707105377417, −1.53603653617730186155043298715, −1.26862769368068891805563630335, −0.68119417147713322337564506440,
0.68119417147713322337564506440, 1.26862769368068891805563630335, 1.53603653617730186155043298715, 1.60979645602740736707105377417, 2.38832022748634222207639775930, 2.51335733051674113947713163323, 2.68248220446785032416315016012, 3.12063464619350709198087589378, 3.25472605722643107306598150096, 3.53470374117965125402303784524, 3.61448118192798378442395823001, 4.04833588597752672251228191017, 4.57671531659987929678886950681, 4.66686379699381178383873813382, 4.68165531553386201126335297889, 4.75180123181174105061516764535, 5.32969437251898836552495153547, 5.34020912141235155753531105717, 5.58625318363115348573890864929, 5.84504045857925779397450540147, 6.26068990860901136944665667563, 6.40577597314999275202493327753, 6.51550777352264895345966044221, 6.69497054129257655538977364344, 6.89078908662822911633680426017
