| L(s) = 1 | + 2·3-s − 16·5-s + 24·7-s − 23·9-s − 62·11-s + 62·13-s − 32·15-s − 17·17-s + 20·19-s + 48·21-s − 12·23-s + 131·25-s − 100·27-s − 80·29-s − 208·31-s − 124·33-s − 384·35-s + 356·37-s + 124·39-s + 22·41-s + 312·43-s + 368·45-s + 24·47-s + 233·49-s − 34·51-s + 462·53-s + 992·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 1.43·5-s + 1.29·7-s − 0.851·9-s − 1.69·11-s + 1.32·13-s − 0.550·15-s − 0.242·17-s + 0.241·19-s + 0.498·21-s − 0.108·23-s + 1.04·25-s − 0.712·27-s − 0.512·29-s − 1.20·31-s − 0.654·33-s − 1.85·35-s + 1.58·37-s + 0.509·39-s + 0.0838·41-s + 1.10·43-s + 1.21·45-s + 0.0744·47-s + 0.679·49-s − 0.0933·51-s + 1.19·53-s + 2.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.490325725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.490325725\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + p T \) |
| good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 + 16 T + p^{3} T^{2} \) |
| 7 | \( 1 - 24 T + p^{3} T^{2} \) |
| 11 | \( 1 + 62 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 12 T + p^{3} T^{2} \) |
| 29 | \( 1 + 80 T + p^{3} T^{2} \) |
| 31 | \( 1 + 208 T + p^{3} T^{2} \) |
| 37 | \( 1 - 356 T + p^{3} T^{2} \) |
| 41 | \( 1 - 22 T + p^{3} T^{2} \) |
| 43 | \( 1 - 312 T + p^{3} T^{2} \) |
| 47 | \( 1 - 24 T + p^{3} T^{2} \) |
| 53 | \( 1 - 462 T + p^{3} T^{2} \) |
| 59 | \( 1 + 240 T + p^{3} T^{2} \) |
| 61 | \( 1 + 812 T + p^{3} T^{2} \) |
| 67 | \( 1 - 216 T + p^{3} T^{2} \) |
| 71 | \( 1 - 732 T + p^{3} T^{2} \) |
| 73 | \( 1 - 178 T + p^{3} T^{2} \) |
| 79 | \( 1 - 700 T + p^{3} T^{2} \) |
| 83 | \( 1 - 992 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 146 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.186395909850777071470375446497, −8.409671174150476273785967191719, −7.85252795078306832019687594847, −7.53426712785951372768854395858, −5.94113007481357650331829356451, −5.11219987900599853993166568937, −4.15093018117655033203409159106, −3.26966489132497029003741412580, −2.17218357054417765141355581396, −0.61199837625277150403082043870,
0.61199837625277150403082043870, 2.17218357054417765141355581396, 3.26966489132497029003741412580, 4.15093018117655033203409159106, 5.11219987900599853993166568937, 5.94113007481357650331829356451, 7.53426712785951372768854395858, 7.85252795078306832019687594847, 8.409671174150476273785967191719, 9.186395909850777071470375446497
