| L(s) = 1 | − 4-s − 9-s + 8·11-s + 16-s + 8·19-s − 4·29-s − 8·31-s + 36-s − 20·41-s − 8·44-s + 14·49-s + 16·59-s + 20·61-s − 64-s + 16·71-s − 8·76-s − 8·79-s + 81-s + 28·89-s − 8·99-s − 12·101-s + 28·109-s + 4·116-s + 26·121-s + 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 2.41·11-s + 1/4·16-s + 1.83·19-s − 0.742·29-s − 1.43·31-s + 1/6·36-s − 3.12·41-s − 1.20·44-s + 2·49-s + 2.08·59-s + 2.56·61-s − 1/8·64-s + 1.89·71-s − 0.917·76-s − 0.900·79-s + 1/9·81-s + 2.96·89-s − 0.804·99-s − 1.19·101-s + 2.68·109-s + 0.371·116-s + 2.36·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.832022114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.832022114\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.102686175362718376924115419268, −8.674976872205014579610399513176, −8.561070558430045159539506325686, −8.035609656333998927155692578483, −7.42637657357619837608742266578, −7.17973435817003348097957125423, −6.68157484600785741437958656523, −6.67711219316901086933526183224, −5.84623157877719780983481316184, −5.59984040164011358027007334065, −5.15512757928069106842408712785, −4.93038821072975749988228583327, −4.05414912395753202368217812084, −3.86206364322777521341628953009, −3.49275402817289348986607960888, −3.23934968937428611353574434117, −2.20686752998281625449359413655, −1.86874871860044182921554026911, −1.11452878732348489724099533534, −0.66659293232709081740859685894,
0.66659293232709081740859685894, 1.11452878732348489724099533534, 1.86874871860044182921554026911, 2.20686752998281625449359413655, 3.23934968937428611353574434117, 3.49275402817289348986607960888, 3.86206364322777521341628953009, 4.05414912395753202368217812084, 4.93038821072975749988228583327, 5.15512757928069106842408712785, 5.59984040164011358027007334065, 5.84623157877719780983481316184, 6.67711219316901086933526183224, 6.68157484600785741437958656523, 7.17973435817003348097957125423, 7.42637657357619837608742266578, 8.035609656333998927155692578483, 8.561070558430045159539506325686, 8.674976872205014579610399513176, 9.102686175362718376924115419268
