| L(s) = 1 | − 3·3-s + 5·5-s − 20·7-s + 9·9-s − 56·11-s + 86·13-s − 15·15-s − 106·17-s + 4·19-s + 60·21-s − 136·23-s + 25·25-s − 27·27-s + 206·29-s + 152·31-s + 168·33-s − 100·35-s − 282·37-s − 258·39-s − 246·41-s + 412·43-s + 45·45-s − 40·47-s + 57·49-s + 318·51-s + 126·53-s − 280·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.07·7-s + 1/3·9-s − 1.53·11-s + 1.83·13-s − 0.258·15-s − 1.51·17-s + 0.0482·19-s + 0.623·21-s − 1.23·23-s + 1/5·25-s − 0.192·27-s + 1.31·29-s + 0.880·31-s + 0.886·33-s − 0.482·35-s − 1.25·37-s − 1.05·39-s − 0.937·41-s + 1.46·43-s + 0.149·45-s − 0.124·47-s + 0.166·49-s + 0.873·51-s + 0.326·53-s − 0.686·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.119263626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.119263626\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + 56 T + p^{3} T^{2} \) |
| 13 | \( 1 - 86 T + p^{3} T^{2} \) |
| 17 | \( 1 + 106 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 136 T + p^{3} T^{2} \) |
| 29 | \( 1 - 206 T + p^{3} T^{2} \) |
| 31 | \( 1 - 152 T + p^{3} T^{2} \) |
| 37 | \( 1 + 282 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 40 T + p^{3} T^{2} \) |
| 53 | \( 1 - 126 T + p^{3} T^{2} \) |
| 59 | \( 1 - 56 T + p^{3} T^{2} \) |
| 61 | \( 1 - 2 T + p^{3} T^{2} \) |
| 67 | \( 1 + 388 T + p^{3} T^{2} \) |
| 71 | \( 1 - 672 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1170 T + p^{3} T^{2} \) |
| 79 | \( 1 + 408 T + p^{3} T^{2} \) |
| 83 | \( 1 - 668 T + p^{3} T^{2} \) |
| 89 | \( 1 - 66 T + p^{3} T^{2} \) |
| 97 | \( 1 + 926 T + p^{3} T^{2} \) |
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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
−9.840165869656113900242108330679, −8.780433855590852940848646976830, −8.097421114944634230891736022617, −6.73866858888088326338016793969, −6.27949137795096327554088179549, −5.47290449320686859309662800371, −4.37106354848871932883604818187, −3.24188000628599877559073452225, −2.11620317809945204600237502109, −0.56486487503757291429927248403,
0.56486487503757291429927248403, 2.11620317809945204600237502109, 3.24188000628599877559073452225, 4.37106354848871932883604818187, 5.47290449320686859309662800371, 6.27949137795096327554088179549, 6.73866858888088326338016793969, 8.097421114944634230891736022617, 8.780433855590852940848646976830, 9.840165869656113900242108330679
