| L(s) = 1 | + 12·5-s + 40·13-s − 16·17-s − 142·25-s − 92·29-s − 328·37-s − 624·41-s − 238·49-s + 532·53-s − 264·61-s + 480·65-s + 492·73-s − 192·85-s − 2.78e3·89-s − 604·97-s + 2.93e3·101-s − 3.12e3·109-s + 3.10e3·113-s − 870·121-s − 3.63e3·125-s + 127-s + 131-s + 137-s + 139-s − 1.10e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.07·5-s + 0.853·13-s − 0.228·17-s − 1.13·25-s − 0.589·29-s − 1.45·37-s − 2.37·41-s − 0.693·49-s + 1.37·53-s − 0.554·61-s + 0.915·65-s + 0.788·73-s − 0.245·85-s − 3.31·89-s − 0.632·97-s + 2.88·101-s − 2.74·109-s + 2.58·113-s − 0.653·121-s − 2.60·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.632·145-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 34 p T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 870 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6550 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4338 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 46 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 59134 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 164 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 312 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 20186 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 178974 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 266 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 346246 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 132 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 343478 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 257070 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 246 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 931870 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 195606 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1392 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 302 T + p^{3} T^{2} )^{2} \) |
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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
−8.454335724385641876241301406715, −8.352853269697175036416224998437, −7.53322603716415626658945077888, −7.40548272531159595479192813363, −6.77919196205150001304123947919, −6.52183601045389796271688640757, −6.09546471330166877581861176464, −5.74306013495796717667003802687, −5.26164519607894616552902495729, −5.13467139041374179437178461999, −4.41536196509713758215398726909, −3.93630733941639700700117854415, −3.49327041316085531818569445788, −3.20871725173744602139833458062, −2.27203965202186773816555075488, −2.17941267894278403263094625231, −1.42295836020293354402691344283, −1.27183521030992646316492767112, 0, 0,
1.27183521030992646316492767112, 1.42295836020293354402691344283, 2.17941267894278403263094625231, 2.27203965202186773816555075488, 3.20871725173744602139833458062, 3.49327041316085531818569445788, 3.93630733941639700700117854415, 4.41536196509713758215398726909, 5.13467139041374179437178461999, 5.26164519607894616552902495729, 5.74306013495796717667003802687, 6.09546471330166877581861176464, 6.52183601045389796271688640757, 6.77919196205150001304123947919, 7.40548272531159595479192813363, 7.53322603716415626658945077888, 8.352853269697175036416224998437, 8.454335724385641876241301406715
