| L(s) = 1 | − 12·3-s − 148·7-s + 63·9-s + 32·13-s + 748·19-s + 1.77e3·21-s + 216·27-s − 2.85e3·31-s − 544·37-s − 384·39-s + 3.87e3·43-s + 1.16e4·49-s − 8.97e3·57-s + 4.46e3·61-s − 9.32e3·63-s + 1.08e4·67-s + 1.07e3·73-s + 1.99e4·79-s − 7.69e3·81-s − 4.73e3·91-s + 3.42e4·93-s + 2.14e4·97-s + 1.84e4·103-s − 1.10e4·109-s + 6.52e3·111-s + 2.01e3·117-s + 1.47e4·121-s + ⋯ |
| L(s) = 1 | − 4/3·3-s − 3.02·7-s + 7/9·9-s + 0.189·13-s + 2.07·19-s + 4.02·21-s + 8/27·27-s − 2.96·31-s − 0.397·37-s − 0.252·39-s + 2.09·43-s + 4.84·49-s − 2.76·57-s + 1.20·61-s − 2.34·63-s + 2.41·67-s + 0.201·73-s + 3.19·79-s − 1.17·81-s − 0.571·91-s + 3.95·93-s + 2.27·97-s + 1.74·103-s − 0.926·109-s + 0.529·111-s + 0.147·117-s + 1.00·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.9408880062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9408880062\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + 4 p T + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 74 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 14702 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 16 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 136622 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 374 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 127502 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 495058 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 46 p T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 272 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5046002 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 1936 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 522418 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14329618 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1036142 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2234 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5416 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 48401282 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 538 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 9962 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 73243022 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 97824962 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10726 T + p^{4} T^{2} )^{2} \) |
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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
−11.16602096555333804933547535671, −10.97950345575480619310245661121, −10.41552820415596158007506923258, −9.868012084983524764217240823515, −9.409474105531697803071871472669, −9.396885377489860925463561786220, −8.724379039211649464506356236979, −7.71176972205542735522211709975, −7.06146021914606700798246527857, −7.06011443195892660010537331460, −6.15468347324386826079764910124, −6.11647581075954075133859192643, −5.36774879344048129163203992573, −5.14909993874118941254089347776, −3.85536483015726984788390841190, −3.56896730254388685394407241499, −3.07778263076803390832483712748, −2.15361183942638066771608837083, −0.70092999563567421087417554016, −0.53775731593148814530779299604,
0.53775731593148814530779299604, 0.70092999563567421087417554016, 2.15361183942638066771608837083, 3.07778263076803390832483712748, 3.56896730254388685394407241499, 3.85536483015726984788390841190, 5.14909993874118941254089347776, 5.36774879344048129163203992573, 6.11647581075954075133859192643, 6.15468347324386826079764910124, 7.06011443195892660010537331460, 7.06146021914606700798246527857, 7.71176972205542735522211709975, 8.724379039211649464506356236979, 9.396885377489860925463561786220, 9.409474105531697803071871472669, 9.868012084983524764217240823515, 10.41552820415596158007506923258, 10.97950345575480619310245661121, 11.16602096555333804933547535671
