L(s) = 1 | − 81·3-s + 2.55e3·5-s + 1.14e4·7-s + 6.56e3·9-s + 1.76e4·11-s + 2.87e3·13-s − 2.06e5·15-s − 4.65e5·17-s − 4.09e5·19-s − 9.30e5·21-s + 3.74e5·23-s + 4.55e6·25-s − 5.31e5·27-s + 3.81e6·29-s + 2.57e6·31-s − 1.43e6·33-s + 2.92e7·35-s + 1.07e7·37-s − 2.33e5·39-s + 1.50e7·41-s + 1.21e7·43-s + 1.67e7·45-s − 4.65e7·47-s + 9.15e7·49-s + 3.76e7·51-s − 1.10e8·53-s + 4.51e7·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.82·5-s + 1.80·7-s + 0.333·9-s + 0.364·11-s + 0.0279·13-s − 1.05·15-s − 1.35·17-s − 0.720·19-s − 1.04·21-s + 0.279·23-s + 2.32·25-s − 0.192·27-s + 1.00·29-s + 0.501·31-s − 0.210·33-s + 3.29·35-s + 0.941·37-s − 0.0161·39-s + 0.831·41-s + 0.540·43-s + 0.608·45-s − 1.39·47-s + 2.26·49-s + 0.779·51-s − 1.92·53-s + 0.664·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(4.007114143\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.007114143\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 81T \) |
good | 5 | \( 1 - 2.55e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.14e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.76e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.87e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.65e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.09e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 3.74e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.81e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.57e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.07e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.50e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.21e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.65e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.10e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.19e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.64e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.68e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.57e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 7.40e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 9.17e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.86e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.12e7T + 7.60e17T^{2} \) |
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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
−9.865386476846028109324400075910, −8.977946597851118371223957070980, −8.097749820654881848388254564507, −6.69305774727702006001489278630, −6.07136003387312722089378991671, −4.99244312008767037811188636567, −4.50293124738445231482510804846, −2.43639688377719806146342068964, −1.75553476823365096176744501935, −0.927787026493126033602394674716,
0.927787026493126033602394674716, 1.75553476823365096176744501935, 2.43639688377719806146342068964, 4.50293124738445231482510804846, 4.99244312008767037811188636567, 6.07136003387312722089378991671, 6.69305774727702006001489278630, 8.097749820654881848388254564507, 8.977946597851118371223957070980, 9.865386476846028109324400075910