| L(s) = 1 | − 2·2-s − 2·4-s − 10·5-s + 20·10-s − 66·11-s + 10·13-s − 20·16-s − 70·17-s + 140·19-s + 20·20-s + 132·22-s + 16·23-s + 75·25-s − 20·26-s + 258·29-s − 20·31-s + 200·32-s + 140·34-s + 328·37-s − 280·38-s − 300·41-s − 116·43-s + 132·44-s − 32·46-s + 30·47-s − 150·50-s − 20·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/4·4-s − 0.894·5-s + 0.632·10-s − 1.80·11-s + 0.213·13-s − 0.312·16-s − 0.998·17-s + 1.69·19-s + 0.223·20-s + 1.27·22-s + 0.145·23-s + 3/5·25-s − 0.150·26-s + 1.65·29-s − 0.115·31-s + 1.10·32-s + 0.706·34-s + 1.45·37-s − 1.19·38-s − 1.14·41-s − 0.411·43-s + 0.452·44-s − 0.102·46-s + 0.0931·47-s − 0.424·50-s − 0.0533·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5674789659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5674789659\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 p T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 p T + 325 p T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 10 T + 19 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 70 T + 6651 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 140 T + 14218 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 16 T + 15774 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 258 T + 40075 T^{2} - 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 20 T + 20082 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 328 T + 106906 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 300 T + 50342 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 116 T + 160794 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 30 T + 190271 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 540 T + 212254 T^{2} + 540 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 380 T + 429258 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1080 T + 705962 T^{2} - 1080 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 468 T + 554906 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1056 T + 949550 T^{2} - 1056 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 860 T + 522934 T^{2} + 860 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 p T - 339825 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 40 T + 703974 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 240 T - 164062 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1630 T + 2133171 T^{2} - 1630 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.771467611427540362011245752386, −8.488961836633519815974487581540, −8.052355647212777719591672763677, −7.954346306536448158441352971162, −7.53661487272273927675004367515, −7.02344270068642938161531517505, −6.67589920278728255573896543101, −6.28093293821484073503299374379, −5.74637737247593893831252281694, −5.11600439857610325225680719294, −4.85891926897435628495308125875, −4.72478003916109560760391500051, −3.95134973313219901378139161800, −3.54681614247695077344597477653, −2.94948912226666004718973292477, −2.65715841504679139586836032888, −2.18604365952443284349527720437, −1.24268827967802971191827004604, −0.74089855284323588550253449091, −0.25803942653553343656550071592,
0.25803942653553343656550071592, 0.74089855284323588550253449091, 1.24268827967802971191827004604, 2.18604365952443284349527720437, 2.65715841504679139586836032888, 2.94948912226666004718973292477, 3.54681614247695077344597477653, 3.95134973313219901378139161800, 4.72478003916109560760391500051, 4.85891926897435628495308125875, 5.11600439857610325225680719294, 5.74637737247593893831252281694, 6.28093293821484073503299374379, 6.67589920278728255573896543101, 7.02344270068642938161531517505, 7.53661487272273927675004367515, 7.954346306536448158441352971162, 8.052355647212777719591672763677, 8.488961836633519815974487581540, 8.771467611427540362011245752386
