| L(s) = 1 | − 3·3-s + 10.4·5-s + 6.42·7-s + 9·9-s − 61.6·11-s + 64.8·13-s − 31.2·15-s − 75.6·17-s + 10.3·19-s − 19.2·21-s − 156.·23-s − 16.3·25-s − 27·27-s − 53.7·29-s + 227.·31-s + 185.·33-s + 66.9·35-s + 10.3·37-s − 194.·39-s + 70.4·41-s − 298.·43-s + 93.7·45-s − 89.9·47-s − 301.·49-s + 227.·51-s + 388.·53-s − 642.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.932·5-s + 0.346·7-s + 0.333·9-s − 1.69·11-s + 1.38·13-s − 0.538·15-s − 1.07·17-s + 0.124·19-s − 0.200·21-s − 1.42·23-s − 0.131·25-s − 0.192·27-s − 0.344·29-s + 1.31·31-s + 0.976·33-s + 0.323·35-s + 0.0458·37-s − 0.798·39-s + 0.268·41-s − 1.05·43-s + 0.310·45-s − 0.279·47-s − 0.879·49-s + 0.623·51-s + 1.00·53-s − 1.57·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| good | 5 | \( 1 - 10.4T + 125T^{2} \) |
| 7 | \( 1 - 6.42T + 343T^{2} \) |
| 11 | \( 1 + 61.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 75.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 53.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 227.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 10.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 70.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 298.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 89.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 388.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 324T + 2.05e5T^{2} \) |
| 61 | \( 1 - 324T + 2.26e5T^{2} \) |
| 67 | \( 1 - 920.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 995.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 362.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.09e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 791.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 150.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.87e3T + 9.12e5T^{2} \) |
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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
−9.759808236985410230177181781195, −8.534866204890713709838693015678, −7.912192020414814435349601215298, −6.62579326908117151843684088880, −5.89567119012448387212568069366, −5.19381106996727299346845073070, −4.11710082384153050518921276684, −2.60043023523854513975889096301, −1.56969571282059101897698431212, 0,
1.56969571282059101897698431212, 2.60043023523854513975889096301, 4.11710082384153050518921276684, 5.19381106996727299346845073070, 5.89567119012448387212568069366, 6.62579326908117151843684088880, 7.912192020414814435349601215298, 8.534866204890713709838693015678, 9.759808236985410230177181781195
