L(s) = 1 | − 2·3-s + 3·4-s − 8·7-s + 3·9-s − 6·12-s + 5·16-s + 8·17-s + 8·19-s + 16·21-s + 8·23-s − 4·27-s − 24·28-s + 9·36-s + 16·37-s − 10·48-s + 34·49-s − 16·51-s − 16·57-s + 24·59-s − 24·63-s + 3·64-s + 24·68-s − 16·69-s + 24·76-s + 5·81-s + 48·84-s + 20·89-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 3/2·4-s − 3.02·7-s + 9-s − 1.73·12-s + 5/4·16-s + 1.94·17-s + 1.83·19-s + 3.49·21-s + 1.66·23-s − 0.769·27-s − 4.53·28-s + 3/2·36-s + 2.63·37-s − 1.44·48-s + 34/7·49-s − 2.24·51-s − 2.11·57-s + 3.12·59-s − 3.02·63-s + 3/8·64-s + 2.91·68-s − 1.92·69-s + 2.75·76-s + 5/9·81-s + 5.23·84-s + 2.11·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1625625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.987359232\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.987359232\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.894946523783105116305876291933, −9.626820318860676955253041725028, −9.386088437362252075495074878424, −8.885705664535840645517521854953, −7.957700128672090165162646483262, −7.59690116380178581010689529032, −7.16122724154490459956563611439, −7.02740253551514500392835680108, −6.36279836599149462874235612059, −6.35046950981171294541604373104, −5.84776591607594587985636413968, −5.42323179478601760960811715972, −5.14049516851820470528905617486, −4.16893951858566354955405928345, −3.45681486625946942967903896636, −3.27238535435631751762886086508, −2.88433016776409213100842773246, −2.30506904182871039880750210074, −0.912227621088752742803498282175, −0.899664157972691799955202863838,
0.899664157972691799955202863838, 0.912227621088752742803498282175, 2.30506904182871039880750210074, 2.88433016776409213100842773246, 3.27238535435631751762886086508, 3.45681486625946942967903896636, 4.16893951858566354955405928345, 5.14049516851820470528905617486, 5.42323179478601760960811715972, 5.84776591607594587985636413968, 6.35046950981171294541604373104, 6.36279836599149462874235612059, 7.02740253551514500392835680108, 7.16122724154490459956563611439, 7.59690116380178581010689529032, 7.957700128672090165162646483262, 8.885705664535840645517521854953, 9.386088437362252075495074878424, 9.626820318860676955253041725028, 9.894946523783105116305876291933