| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.003582201669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003582201669\) |
| \(L(1)\) |
|
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
−7.46376327468835445454077251214, −6.95256808480429624874253422336, −6.81862783948448530626835031616, −6.59312752757852423803732211797, −6.28261202652071340789819072997, −5.74770087940335500721356019633, −5.65059145788940228130233170726, −5.59691318252514024965717626508, −5.50668732918627358443175295474, −5.17964953114655331712123517873, −5.08220859522155468642935764826, −4.76844259961484269743955896908, −4.55864405236426894536662985868, −4.32454306717423570288633879372, −4.14974736853597503048552457805, −3.53661093846183708486206301997, −3.29696845915797981696393607920, −3.25896897379239515047641494336, −3.11464624283162556617639222986, −2.47158519213021804123788461998, −2.42132954448315411665851392140, −1.99781546648243450477719369885, −1.94688066311879042378778691487, −0.885842366840783675584110855859, −0.05280534375367896970166015406,
0.05280534375367896970166015406, 0.885842366840783675584110855859, 1.94688066311879042378778691487, 1.99781546648243450477719369885, 2.42132954448315411665851392140, 2.47158519213021804123788461998, 3.11464624283162556617639222986, 3.25896897379239515047641494336, 3.29696845915797981696393607920, 3.53661093846183708486206301997, 4.14974736853597503048552457805, 4.32454306717423570288633879372, 4.55864405236426894536662985868, 4.76844259961484269743955896908, 5.08220859522155468642935764826, 5.17964953114655331712123517873, 5.50668732918627358443175295474, 5.59691318252514024965717626508, 5.65059145788940228130233170726, 5.74770087940335500721356019633, 6.28261202652071340789819072997, 6.59312752757852423803732211797, 6.81862783948448530626835031616, 6.95256808480429624874253422336, 7.46376327468835445454077251214
