| L(s) = 1 | + 9·3-s + 108.·5-s + 213.·7-s + 81·9-s − 401.·11-s − 169·13-s + 973.·15-s − 1.70e3·17-s + 373.·19-s + 1.92e3·21-s + 2.35e3·23-s + 8.56e3·25-s + 729·27-s − 2.93e3·29-s + 2.06e3·31-s − 3.60e3·33-s + 2.30e4·35-s + 1.39e4·37-s − 1.52e3·39-s + 3.62e3·41-s + 1.81e4·43-s + 8.75e3·45-s − 1.18e4·47-s + 2.87e4·49-s − 1.53e4·51-s − 1.34e4·53-s − 4.33e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.93·5-s + 1.64·7-s + 0.333·9-s − 0.999·11-s − 0.277·13-s + 1.11·15-s − 1.43·17-s + 0.237·19-s + 0.950·21-s + 0.926·23-s + 2.74·25-s + 0.192·27-s − 0.647·29-s + 0.385·31-s − 0.576·33-s + 3.18·35-s + 1.67·37-s − 0.160·39-s + 0.337·41-s + 1.49·43-s + 0.644·45-s − 0.780·47-s + 1.71·49-s − 0.827·51-s − 0.656·53-s − 1.93·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.015835463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.015835463\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 13 | \( 1 + 169T \) |
| good | 5 | \( 1 - 108.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 213.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 401.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.70e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 373.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.35e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.06e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.62e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.81e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.18e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.34e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.62e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.03e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.41e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.06e5T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.81e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.22e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.24e5T + 8.58e9T^{2} \) |
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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
−9.678898849685064217966673390695, −9.061704465305056461415882129486, −8.189248379587232770798656981448, −7.27456146439643197788361532711, −6.12491152672947748266161869564, −5.16606387490170922762089770562, −4.57653853870386938243487044266, −2.61482769082547122413051815636, −2.14040649820841515340816500108, −1.13800346869777333813957498659,
1.13800346869777333813957498659, 2.14040649820841515340816500108, 2.61482769082547122413051815636, 4.57653853870386938243487044266, 5.16606387490170922762089770562, 6.12491152672947748266161869564, 7.27456146439643197788361532711, 8.189248379587232770798656981448, 9.061704465305056461415882129486, 9.678898849685064217966673390695
