| L(s) = 1 | − 2·3-s + 30·5-s − 38·7-s + 2·9-s − 404·11-s + 198·13-s − 60·15-s − 478·17-s + 76·21-s + 1.08e3·23-s + 275·25-s − 162·27-s − 1.51e3·31-s + 808·33-s − 1.14e3·35-s − 282·37-s − 396·39-s + 2.08e3·41-s + 1.51e3·43-s + 60·45-s − 918·47-s + 722·49-s + 956·51-s + 3.63e3·53-s − 1.21e4·55-s − 4.16e3·61-s − 76·63-s + ⋯ |
| L(s) = 1 | − 2/9·3-s + 6/5·5-s − 0.775·7-s + 2/81·9-s − 3.33·11-s + 1.17·13-s − 0.266·15-s − 1.65·17-s + 0.172·21-s + 2.04·23-s + 0.439·25-s − 2/9·27-s − 1.57·31-s + 0.741·33-s − 0.930·35-s − 0.205·37-s − 0.260·39-s + 1.23·41-s + 0.820·43-s + 0.0296·45-s − 0.415·47-s + 0.300·49-s + 0.367·51-s + 1.29·53-s − 4.00·55-s − 1.11·61-s − 0.0191·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3822533615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3822533615\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 6 p T + p^{4} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p^{4} T^{3} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 38 T + 722 T^{2} + 38 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 202 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 198 T + 19602 T^{2} - 198 p^{4} T^{3} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 478 T + 114242 T^{2} + 478 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 259042 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 1082 T + 585362 T^{2} - 1082 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1374562 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 758 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 282 T + 39762 T^{2} + 282 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 1042 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1518 T + 1152162 T^{2} - 1518 p^{4} T^{3} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 918 T + 421362 T^{2} + 918 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3638 T + 6617522 T^{2} - 3638 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3074722 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2082 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10162 T + 51633122 T^{2} + 10162 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3478 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6958 T + 24206882 T^{2} + 6958 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 18917762 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12162 T + 73957122 T^{2} + 12162 p^{4} T^{3} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 93222082 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 1122 T + 629442 T^{2} - 1122 p^{4} T^{3} + p^{8} T^{4} \) |
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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
−11.28275477479424055847853853758, −10.67311857481992680178335481008, −10.30703949199127766094724608333, −10.11783098346465022192483714039, −9.230647453578072845916513774475, −8.851691385328123441061507893360, −8.740916216063692173824941255754, −7.68208033154787340493698688239, −7.45807702490342026834339261858, −6.90293851792095488394535888661, −6.16975008217713329828674469983, −5.67550787745622993320817432537, −5.53363763890965849943050744972, −4.82174112416020544758767822297, −4.24408240264528996187425838318, −3.11386169572921557001847484686, −2.77013365927301879941275080315, −2.25026753773918079769523674506, −1.36048843328874145612985104599, −0.18370834951006218937582595171,
0.18370834951006218937582595171, 1.36048843328874145612985104599, 2.25026753773918079769523674506, 2.77013365927301879941275080315, 3.11386169572921557001847484686, 4.24408240264528996187425838318, 4.82174112416020544758767822297, 5.53363763890965849943050744972, 5.67550787745622993320817432537, 6.16975008217713329828674469983, 6.90293851792095488394535888661, 7.45807702490342026834339261858, 7.68208033154787340493698688239, 8.740916216063692173824941255754, 8.851691385328123441061507893360, 9.230647453578072845916513774475, 10.11783098346465022192483714039, 10.30703949199127766094724608333, 10.67311857481992680178335481008, 11.28275477479424055847853853758
