| L(s) = 1 | − 32·4-s + 484·7-s + 1.02e3·16-s + 5.28e3·25-s − 1.54e4·28-s + 1.39e4·31-s + 1.42e5·49-s − 3.27e4·64-s − 1.81e5·73-s − 3.41e4·79-s + 1.80e5·97-s − 1.69e5·100-s − 2.65e5·103-s + 4.95e5·112-s − 2.27e5·121-s − 4.44e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.42e5·169-s + 173-s + ⋯ |
| L(s) = 1 | − 4-s + 3.73·7-s + 16-s + 1.69·25-s − 3.73·28-s + 2.59·31-s + 8.45·49-s − 64-s − 3.98·73-s − 0.614·79-s + 1.94·97-s − 1.69·100-s − 2.46·103-s + 3.73·112-s − 1.40·121-s − 2.59·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 2·169-s + 2.54e−6·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.461400677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.461400677\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{5} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 5282 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 242 T + p^{5} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 227050 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 36304750 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6950 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 617985986 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 588563050 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 90706 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 17050 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7877173786 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 90242 T + p^{5} T^{2} )^{2} \) |
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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
−14.38885568612865637135553876118, −13.57245897161940406175792108843, −12.94320265817395867085944612012, −11.95659538454258129545109102653, −11.80609030610155248132742684652, −11.24261365348572579073215171427, −10.41184426592447219899532832691, −10.37770653135890118806023287147, −8.998297255163238349018262213974, −8.714623620003366236881615839631, −8.014055900773954957083966877491, −7.950181939760842696281410646825, −7.01380335435642547359691954800, −5.74336839320840225875864073874, −5.08224454962027302780892910397, −4.56147285955063336232936591773, −4.33740463872943363496984750649, −2.69791306455286110999393500833, −1.47471772346094969711873392253, −1.01412341688442756353742463179,
1.01412341688442756353742463179, 1.47471772346094969711873392253, 2.69791306455286110999393500833, 4.33740463872943363496984750649, 4.56147285955063336232936591773, 5.08224454962027302780892910397, 5.74336839320840225875864073874, 7.01380335435642547359691954800, 7.950181939760842696281410646825, 8.014055900773954957083966877491, 8.714623620003366236881615839631, 8.998297255163238349018262213974, 10.37770653135890118806023287147, 10.41184426592447219899532832691, 11.24261365348572579073215171427, 11.80609030610155248132742684652, 11.95659538454258129545109102653, 12.94320265817395867085944612012, 13.57245897161940406175792108843, 14.38885568612865637135553876118
