| L(s) = 1 | − 4·3-s + 22·9-s − 60·11-s − 22·17-s + 12·19-s − 74·25-s − 128·27-s + 240·33-s − 282·41-s − 80·43-s + 46·49-s + 88·51-s − 48·57-s + 228·59-s − 24·67-s + 296·75-s + 449·81-s + 24·89-s − 336·97-s − 1.32e3·99-s − 200·107-s + 350·113-s + 1.85e3·121-s + 1.12e3·123-s + 127-s + 320·129-s + 131-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 22/9·9-s − 5.45·11-s − 1.29·17-s + 0.631·19-s − 2.95·25-s − 4.74·27-s + 7.27·33-s − 6.87·41-s − 1.86·43-s + 0.938·49-s + 1.72·51-s − 0.842·57-s + 3.86·59-s − 0.358·67-s + 3.94·75-s + 5.54·81-s + 0.269·89-s − 3.46·97-s − 13.3·99-s − 1.86·107-s + 3.09·113-s + 15.3·121-s + 9.17·123-s + 0.00787·127-s + 2.48·129-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06464798117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06464798117\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + p T^{2} + p^{4} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + 2 T - 5 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 37 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 46 T^{2} - 285 T^{4} - 46 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 30 T + 421 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 11 T - 168 T^{2} + 11 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T + 373 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 434 T^{2} - 91485 T^{4} + 434 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^3$ | \( 1 + 1643 T^{2} + 1992168 T^{4} + 1643 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 1714 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 1019 T^{2} - 835800 T^{4} + 1019 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 141 T + 8308 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 40 T - 249 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 1870 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 5267 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 114 T + 7813 T^{2} - 114 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^3$ | \( 1 + 7403 T^{2} + 40958568 T^{4} + 7403 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 12 T + 4537 T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - 1294 T^{2} - 23737245 T^{4} - 1294 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 10631 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 11078 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 7390 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 12 T + 7969 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 168 T + 18817 T^{2} + 168 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.78749871049125621022701476117, −7.75919601045726191064877748081, −7.36031889540309685579846000269, −7.07953200533427400498786104636, −6.91145589776273146438581217101, −6.80863869259217391125650129822, −6.54126512781956620692564033713, −5.74074266501691141564109127044, −5.68717101104218994409482536052, −5.57297499708980581864884176852, −5.47997291821049927414322771302, −5.14135264286985991179257768963, −4.85251664564594288001932260081, −4.65204899768870764121534271685, −4.51252116427572077253013303658, −3.70053515707283808342011779828, −3.62503327434661871831304368285, −3.50395081190536856872207261826, −2.82688971138214751896769710175, −2.41085292856561358663848656627, −2.16429604722401461950851667903, −1.78541903243586882714351974363, −1.65594478662231164432843592864, −0.31355993628236808151463344641, −0.14748382793434221162806569648,
0.14748382793434221162806569648, 0.31355993628236808151463344641, 1.65594478662231164432843592864, 1.78541903243586882714351974363, 2.16429604722401461950851667903, 2.41085292856561358663848656627, 2.82688971138214751896769710175, 3.50395081190536856872207261826, 3.62503327434661871831304368285, 3.70053515707283808342011779828, 4.51252116427572077253013303658, 4.65204899768870764121534271685, 4.85251664564594288001932260081, 5.14135264286985991179257768963, 5.47997291821049927414322771302, 5.57297499708980581864884176852, 5.68717101104218994409482536052, 5.74074266501691141564109127044, 6.54126512781956620692564033713, 6.80863869259217391125650129822, 6.91145589776273146438581217101, 7.07953200533427400498786104636, 7.36031889540309685579846000269, 7.75919601045726191064877748081, 7.78749871049125621022701476117
