| L(s) = 1 | + 2·3-s + 4-s + 3·9-s + 2·12-s − 13-s + 25-s + 4·27-s + 3·36-s − 3·37-s − 2·39-s + 43-s − 52-s − 2·61-s − 64-s − 3·73-s + 2·75-s + 2·79-s + 5·81-s + 3·97-s + 100-s − 103-s + 4·108-s − 3·109-s − 6·111-s − 3·117-s − 2·121-s + 127-s + ⋯ |
| L(s) = 1 | + 2·3-s + 4-s + 3·9-s + 2·12-s − 13-s + 25-s + 4·27-s + 3·36-s − 3·37-s − 2·39-s + 43-s − 52-s − 2·61-s − 64-s − 3·73-s + 2·75-s + 2·79-s + 5·81-s + 3·97-s + 100-s − 103-s + 4·108-s − 3·109-s − 6·111-s − 3·117-s − 2·121-s + 127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3651921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.239776127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.239776127\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.309644733601260581145283972686, −9.174069432301650667015551679929, −8.865997004054478147977700228350, −8.507880347017384678144413303819, −7.945262315351287449566326941441, −7.58856228742665844080815196510, −7.37823322609500111327843976467, −7.01146288541446575869758535933, −6.55349627360528221583225869190, −6.39013395271185543641049770694, −5.46841893645121303954388477802, −5.09144083487460602071559284649, −4.52815690637884119446611512614, −4.23323282315308325054342726737, −3.51205808630915673771221989861, −3.19922399102199989391815765240, −2.71921494366827768598467662107, −2.37718205174090685591041442415, −1.77747162008191393917267290324, −1.39544574902744931859520919945,
1.39544574902744931859520919945, 1.77747162008191393917267290324, 2.37718205174090685591041442415, 2.71921494366827768598467662107, 3.19922399102199989391815765240, 3.51205808630915673771221989861, 4.23323282315308325054342726737, 4.52815690637884119446611512614, 5.09144083487460602071559284649, 5.46841893645121303954388477802, 6.39013395271185543641049770694, 6.55349627360528221583225869190, 7.01146288541446575869758535933, 7.37823322609500111327843976467, 7.58856228742665844080815196510, 7.945262315351287449566326941441, 8.507880347017384678144413303819, 8.865997004054478147977700228350, 9.174069432301650667015551679929, 9.309644733601260581145283972686
